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The  Publishers  and  the  Authors  will  be  grateful  to 
any  of  the  readers  of  this  volume  who  will  kindly  call 
their  attention  to  any  errors  of  omission  or  of  commis- 
sion that  they  may  find  therein.  It  is  intended  to  make 
oui*  publications  standard  works  of  study  and  reference, 
and,  to  that  end,  the  greatest  accuracy  is  sought.  It 
rarely  happens  that  the  early  editions  of  works  of  any 
size  are  free  from  errors;  but  it  is  the  endeavor  of  the 
Publishers  to  have  them  removed  immediately  upon  being 
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may  be  aided  in  their  task  of  revision,  from  time  to  time, 
by  the  kindly  criticism  of  their  readers. 

JOHN  WILEY  &  SONS,  Inc. 
432  4TH  AVENUE. 


WORKS  OF  WILLIAM  KENT 

PUBLISHED   BY 

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The  Mechanical  Engineers  Pocket-Book. 

A  Reference  Book  01  Rules,  Tables,  Data,  and 
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THE 

MECHANICAL  ENGINEERS' 
POCKET-BOOK. 


A  REFERENCE-BOOK  OF  RULES,  TABLES, 
DATA,  AND  FORMULA. 


WILLIAM  KENT,  M.E.,  Sc.D., 

Consulting  Engineer. 
Member  Amer.  Soc'y  Mechl.  Engrs.  and  Amer.  Inst.  Mining  Engrs. 


NINTH  EDITION,  THOROUGHLY  REVISED 

WITH  THE  ASSISTANCE   OF 

ROBERT  THURSTON  KENT,  M.  E., 

Consulting  Engineer. 
Junior  American  Society  of  Mechanical  Engineers. 

TOTAL  I3SJW, 
ONE  HUNDRED  AND  THIRTY-FIVE  THOUSAND. 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 

LONDON:  CHAPMAN  &  HALL,  LIMITED 

1916 


COPYRIGHT,  1895,  1902,  1910,  1915, 

BY 

WILLIAM  KENT. 


Eighth  Edition  entered  at  Stationers'  Hall. 


Composition  and  Electrotyping  by  the  STANHOPE  PRESS,  Boston,  Mass.,  and 

the  PUBLISHERS  PRINTING  COMPANY,  New  York. 
Printing  and  Binding  by  BRAUNWORTH  &  COMPANY,  Brooklyn,  N.  Y. 


PREFACE  TO  THE  NINTH  EDITION. 

NOVEMBER,  1915. 

SINCE  the  eighth  edition  was  published,  five  years  ago,  there  have 
been  notable  advances  in  many  branches  of  engineering,  rendering 
obsolete  portions  of  the  book  which  at  that  time  were  in  accord  with 
practice.  In  addition,  many  engineering  standards  have  been  changed 
during  the  five-year  period,  necessitating  a  thorough  revision  of  many 
sections  of  the  work.  The  absolutely  necessary  revisions  to  bring  the 
book  up  to  date  -have  involved  changes  in  over  400  pages  of  the  eighth 
edition,  and  the  addition  of  over  150  pages  of  new  matter.  The 
treatment  of  many  subjects  in  the  earlier  edition  has  been  condensed 
into  smaller  space  to  enable  the  insertion  of  the  new  matter  without 
increasing  the  size  of  the  book  to  unwieldy  proportions.  Extensive 
revisions  have  been  made  in  the  subjects  of  materials,  mechanics, 
fans  and  blowers,  heating  and  ventilation,  fuel,  steam-boilers  and 
engines,  and  steam-turbines.  The  chapter  on  machine-shop  practice 
has  been  rewritten  and  doubled  in  size,  and  now  covers  many  subjects 
which  were  omitted  in  earlier  editions.  The  new  matter  includes 
many  data  on  planing,  milling,  drilling  and  grinding,  together  with 
an  elaborate  treatment  of  the  subject  of  machine-tool  driving.  The 
subject  of  electrical  engineering  has  been  completely  rewritten  and 
brought  into  agreement  with  present  practice.  Of  the  new  tables 
added  the  following  are  considered  of  special  importance.  Square 
roots  of  fifth  powers;  Four-place  logarithms;  Standard  sizes  of  welded 
steel  pipe;  Standard  pipe  flanges;  Properties  of  wire  rope;  Fire  brick 
and  other  refractories;  Properties  of  structural  sections  and  columns; 
Chemical  standards  for  iron  castings;  Flow  of  air,  water  and  steam; 
Analyses  and  heating  values  of  coals;  Rankine  efficiency;  Cooling 
towers;  Properties  of  ammonia;  -Power  required  for  driving  machine 
tools  of  all  types,  both  singly  and  in  groups;  Electric  resistance  and 
conductivity  of  wires;  Street  railway  installation;  Electric  lamp  char- 
acteristics; Illuminating  data. 

NOTE  TO  SECOND  PRINTING  OF  THE  NINTH 
EDITION. 

In  line  with  the  policy  of  keeping  the  book  up  to  date  and  elimi- 
nating all  obsolete  matter,  the  section  on  hydraulic  turbines  has  been 
completely  rewritten  for  the  second  printing  of  the  ninth  edition. 
The  presentation  of  the  theory  has  been  improved,  new  design  con- 
stants have  been  given,  and  the  tables  of  capacity,  etc.,  represent  the 
performance  of  the  most  recent  types  of  turbines. 

MARCH,  1917. 

iii 


40223,3 


IV  PREFACE. 


ABSTRACT  FROM  PREFACE  TO  THE 
FIRST  EDITION,  1895. 

MORE  than  twenty  years  ago  the  author  began  to  follow  the  advice 
given  by  Nystrom:  "  Every  engineer  should  make  his  own  pocket-book, 
as  he  proceeds  in  study  and  practice,  to  suit  his  particular  business." 
The  manuscript  pocket-book  thus  begun,  however,  soon  gave  place  to 
more  modern  means  for  disposing  of  the  accumulation  of  engineering 
facts  and  figures,  viz.,  the  index  rerum,  the  scrap-book,  the  collection  of 
indexed  envelopes,  portfolios  and  boxes,  the  card  catalogue,  etc.  Four 
years  ago,  at  the  request  of  the  publishers,  the  labor  was  begun  of  selecting 
from  this  accumulated  mass  such  matter  as  pertained  to  mechanical 
engineering,  and  of  condensing,  digesting,  and  arranging  it  in  form  for 
publication.  In  addition  to  this,  a  careful  examination  was  made  of  the 
transactions  of  engineering  societies,  and  of  the  most  important  recent 
works  on  mechanical  engineering,  in  order  to  fill  gaps  that  might  be  left 
in  the  original  collection,  and  insure  that  no  important  facts  had  been 
overlooked. 

Some  ideas  have  been  kept  in  mind  during  the  preparation  of  the 
Pocket-book  that  will,  it  is  believed,  cause  it  to  differ  from  other  works 
of  its  class.  In  the  first  place  it  was  considered  that  the  field  of  mechani- 
cal engineering  was  so  great,  and  the  literature  of  the  subject  so  vast,  that 
as  little  space  as  possible  should  be  given  to  subjects  which  especially 
belong  to  civil  engineering.  While  the  mechanical  engineer  must  con- 
tinually deal  with  problems  which  belong  properly  to  civil  engineering, 
this  latter  branch  is  so  well  covered  by  Traut wine's  "  Civil  Engineer's 
Pocket-book  "  that  any  attempt  to  treat  it  exhaustively  would  not  only 
fill  no  "  long-felt  want,"  but  would  occupy  space  which  should  be  given 
to  mechanical  engineering. 

Another  idea  prominently  kept  in  view  by  the  author  has  been  that  he 
would  not  assume  the  position  of  an  "  authority  "  in  giving  rules  and 
formulae  for  designing,  but  only  that  of  compiler,  giving  not  only  the 
name  of  the  originator  of  the  rule,  where  it  was  known,  but  also  the  volume 
and  page  from  which  it  was  taken,  so  that  its  derivation  may  be  traced 
when  desired.  When  different  formulas  for  the  same  problem  have  been 
found  they  have  been  given  in  contrast,  and  in  many  cases  examples 
have  been  calculated  by  each  to  show  the  difference  between  them.  In 
some  cases  these  differences  are  quite  remarkable,  as  will  be  seen  under 
Safety-valves  and  Crank-pins.  Occasionally  the  study  of  these  differences 
has  led  to  the  author's  devising  a  new  formula,  in  which  case  the  deriva- 
tion of  the  formula  is  given. 

Much  attention  has  been  paid  to  the  abstracting  of  data  of  experiments 
from  recent  periodical  literature,  and  numerous  references  to  other  data 
are  given.  In  this  respect  the  present  work  will  be  found  to  differ  from 
other  Pocket-books. 

The  author  desires  to  express  his  obligation  to  the  many  persons  who 
huve  assisted  him  in  the  preparation  of  the  work,  to  manufacturers  who 


PREFACE.  V 

have  furnished  their  catalogues  and  given  permission  for  the  use  of  their 
tables,  and  to  many  engineers  who  have  contributed  original  data  and 
tables.  The  names  of  these  persons  are  mentioned  in  their  proper  places 
in  the  text,  and  in  all  cases  it  has  been  endeavored  to  give  credit  to  whom 
credit  is  due. 

WILLIAM  KENT. 


PREFACE  TO  THE  EIGHTH  EDITION. 

SEPTEMBER,  1910. 

DURING  the  first  ten  years  following  the  issue  of  the  first  edition  of  this 
book,  in  1895,  the  attempt  was  made  to  keep  it  up  to  date  by  the  method 
of  cutting  out  pages  and  paragraphs,  inserting  new  ones  in  their  places,  by 
inserting  new  pages  lettered  a,  b,  c,  etc.,  and  by  putting  some  new  matter 
in  an  appendix.  In  this  way  the  book  passed  to  its  7th  edition  in  October, 
1904.  After  50,000  copies  had  been  printed  it  was  found  that  the  electro- 
typed  plates  were  beginning  to  wear  out,  so  that  extensive  resetting  of  type 
would  soon  be  necessary.  The  advances  in  engineering  practice  also  had 
been  so  great  that  it  was  evident  that  many  chapters  required  to  be  entirely 
rewritten.  It  was  therefore  determined  to  make  a  thorough  revision  of  the 
book,  and  to  reset  the  type  throughout.  This  has  now  been  accomplished 
after  four  years  of  hard  labor.  The  size  of  the  book  has  increased  over  300 
pages,  in  spite  of  all  efforts  to  save  space  by  condensation  and  elision  of 
much  of  the  old  matter  and  by  resetting  many  of  the  tables  and  formulae 
in  shorter  form.  A  new  style  of  type  for  the  tables  has  been  designed  for 
the  book,  which  is  believed  to  be  much  more  easily  read  than  the  old. 

The  thanks  of  the  author  are  due  to  many  manufacturers  who  ha^re  fur- 
nished new  tables  of  materials  and  machines,  and  to  many  engineers  who 
have  made  valuable  contributions  and  helpful  suggestions.  He  is  especially 
indebted  to  his  son,  Robert  Thurston  Kent,  M.E.,  who  has  done  the  work 
of  revising  manufacturers'  tables  of  materials  and  has  done  practically  all 
of  the  revising  of  the  subjects  of  Compressed  Air,  Fans  and  Blowers,  Hoist- 
ing  and  Conveying,  and  Machine  Shop. 


CONTENTS. 

(For  Alphabetical  Index  see  page  1479.) 

MATHEMATICS. 

Arithmetic. 

PAGE 

Arithmetical  and  Algebraical  Signs 1 

Greatest  Common  Divisor 2 

Least  Common  Multiple 

Fractions 

Decimals 

Table.     Decimal  Equivalents  of  Fractions  of  One  Inch 3 

Table.     Products  of  Fractions  expressed  in  Decimals 

Compound  or  Denominate  Numbers 5 

Reduction  Descending  and  Ascending 5 

Decimals  of  a  Foot  Equivalent  to  Fractions  of  an  Inch 5 

Ratio  and  Proportion 6 

Involution,  or  Powers  of  Numbers 7 

Table.     First  Nine  Powers  of  the  First  Nine  Numbers 7 

Table.     First  Forty  Powers  of  2 8 

Evolution.     Square  Root 8 

Cube  Root 9 

Alligation 9 

Permutation 10 

Combination 10 

Arithmetical  Progression 10 

Geometrical  Progression 11 

Percentage,  Profit  and  Loss,  Efficiency 12 

Interest 12 

Discount 13 

Compound  Interest 

Compound  Interest  Table,  3,  4,  5,  and  6  per  cent 

Equation  of  Payments >, 

Partial  Payments 14 

Annuities 15 

Tables  of  Amount,  Present  Values,  etc.,  of  Annuities 15 

Weights  and  Measures. 

Long  Measure 17 

Old  Land  Measure 17 

Nautical  Measure 17 

Square  Measure 

Solid  or  Cubic  Measure 

Liquid  Measure 

The  Miners'  Inch 

Apothecaries'  Fluid  Measure 

Dry  Measure i ~ 19 

Shipping  Measure 

Avoirdupois  Weight 19 

Troy  Weight 19 

Apothecaries'  Weight 20 

To  Weigh  Correctly  on  an  Incorrect  Balance 20 

Circular  Measure 20 

Measure  of  Time 20 

vii 


Vlll  CONTENTS. 

PAGE 

Board  and  Timber  Measure 20 

Table.     Contents  in  Feet  of  Joists,  Scantlings,  and  Timber.  ...  21 

French  or  Metric  Measures 21 

British  and  French  Equivalents 22 

Metric  Conversion  Tables 23 

Compound  Units 

of  Pressure  and  Weight 27 

of  Water,  Weight  and  Bulk 27 

of  Air,  Weight  and  Volume 27 

of  Work,  Power,  and  Duty .  27 

of  Velocity 4 27 

Wire  and  Sheet  Metal  Gages 28 

Circular-mil  Wire  Gage 29,  30 

U.  S.  Standard  Wire  and  Sheet  Gage  (1893) 29,  32 

Twist-drill  and  Steel-wire  Gages 31 

Decimal  Gage 32 

Algebra. 

Addition,  Multiplication,  etc 33 

Powers  of  Numbers 

Parentheses,  Division 

Simple  Equations  and  Problems 

Equations  containing  two  or  more  Unknown  Quantities 

Elimination 

Quadratic  Equations 

Theory  of  Exponents 

Binominal  Theorem 

Geometrical  Problems  of  Construction 

of  Straight  Lines 37 

^f  Angles 38 

of  Circles 39 

of  Triangles 

of  Squares  and  Polygons 

of  the  Ellipse 45 

of  the  Parabola 

of  the  Hyperbola 

of  the  Cycloid 50 

of  the  Tractrix  or  Schiele  Anti-friction  Curve 50 

of  the  Spiral 51 

of  Rings  inside  a  Circle '51 

of  Arc  of  a  Large  Circle 51 

of  the  Catenary 52 

of  the  Involute 52 

of  plotting  Angles 

Geometrical  Propositions 53 

Degree  of  a  Railway  Curve 54 

Mensuration,  Plane  Surfaces. 

Quadrilateral,  Parallelogram,  etc 54 

Trapezium  and  Trapezoid 54 

Triangles 54 

Polygons.     Table  of  Polygons 55 

Irregular  Figures 56 

Properties  of  the  Circle 57 

Values  of  TT  and  its  Multiples,  etc 57 

Relations  of  arc,  chord,  etc 58 

Relations  of  circle  to  inscribed  square,  etc 59 

Formulse  for  a  Circular  Curve 59 

Sectors  and  Segments 60 

Circular  Ring 60 

The  Ellipse 60 

The  Helix 61 

The  Spiral 61 

Surfaces  and  Volumes  of  Similar  Solids 61 


CONTENTS.  ix 

Mensuration,  Solid  Bodies.  PAGE 

Prism 62 

Pyramid 62 

Wedge 62 

Rectangular  Prismoid 62 

Cylinder 62 

Cone 62 

Sphere 62 

Spherical  Triangle 63 

Spherical  Polygon 63 

The  Prismoid 63 

The  Prismoidal  Formula 63 

Polyedron 63 

Spherical  Zone 64 

Spherical  Segment 64 

Spheroid  or  Ellipsoid 64 

Cylindrical  Ring ; 64 

Solids  of  Revolution 64 

Spindles 64 

Frustum  of  a  Spheroid 64 

Parabolic  Conoid 65 

Volume  of  a  Cask 65 

'Irregular  Solids 65 

Plane  Trigonometry. 

Solution  of  Plane  Triangles 66 

Sine,  Tangent,  Secant,  etc ...'..' 66 

Signs  of  the  Trigonometric  Functions 67 

Trigonometrical  Formulae 68 

Solution  of  Plane  Right-angled  Triangles 69 

Solution  of  Oblique-angled  Triangles 69 

Analytical  Geometry. 

Ordinates  and  Abscissas 70 

Equations  of  a  Straight  Line,  Intersections,  etc 70 

Equations  of  the  Circle 71 

Equations  of  the  Ellipse 71 

Equations  of  the  Parabola 72 

Equations  of  the  Hyperbola 72 

Logarithmic  Curves 73 

Differential  Calculus. 

Definitions 73 

Differentials  of  Algebraic  Functions 74 

Formulae  for  Differentiating 74 

Partial  Differentials 75 

Integrals 75 

Formulae  for  Integration 75 

Integration  between  Limits 76 

Quadrature  of  a  Plane  Surface 76 

Quadrature  of  Surfaces  of  Revolution 77 

Cubature  of  Volumes  of  Revolution ' 77 

Second,  Third,  etc.,  Differentials 77 

Maclaurin's  and  Taylor's  Theorems 78 

Maxima  arid  Minima 78 

Differential  of  an  Exponential  Function 79 

Logarithms 79 

Differential  Forms  which  have  Known  Integrals 80 

Exponential  Functions .  . 80 

Circular  Functions 81 

The  Cycloid 81 

Integral  Calculus 82 


X  CONTENTS. 

The  Slide  Bule. 

Examples  solved  by  the  Slide  Rule .  .  . 82 

Logarithmic  Ruled  Paper. 

Plotting  on  Logarithmic  Paper 84 

Mathematical  Tables. 

Formula  for  Interpolation 86 

Reciprocals  of  Numbers  1  to  2000 

Squares,  Cubes,  Square  Roots  and  Cube  Roots  from  0.1  to  1600 

Squares  and  Cubes  of  Decimals 108 

Fifth  Roots  and  Fifth  Powers 109 

Square  Roots  of  Fifth  Powers  of  Pipe  Sizes 

Circumferences  and  Areas  of  Circles Ill 

Circumferences  of  Circles  in  Feet  and  Inches  from  1  inch  to  32 

feet  11  inches  in  diameter 

Areas  of  the  Segments  of  a  Circle 

Lengths  of  Circular  Arcs,  Degrees  Given 

Lengths  of  Circular  Arcs,  Height  of  Arc  Given 

Circles  and  Squares  of  Equal  Area 125 

Number  of  Circles  Inscribed  within  a  Large  Circle 125 

Spheres 126 

Square  Feet  in  Plates  3  to  32  feet  long  and  1  inch  wide 128 

Gallons  in  a  Number  of  Cubic  Feet 

Cubic  Feet  in  a  Number  of  Gallons 130 

Contents  of  Pipes  and  Cylinders,  Cubic  Feet  and  Gallons 

Cylindrical  Vessels,  Tanks,  Cisterns,  etc 132 

Capacities  of  Rectangular  Tanks  in  Gallons 

Number  of  Barrels  in  Cylindrical  Cisterns  and  Tanks 

Logarithms 135 

Table  of  Logarithms 

Hyperbolic  Logarithms 

Four-place  Logarithms  of  Numbers  from  1  to  1000 167 

Natural  Trigonometric  Functions 169 

Logarithmic  Trigonometric  Functions 172 

MATERIALS. 

Chemical  Elements 173 

Specific  Gravity  and  Weight  of  Materials 173 

The  Hydrometer : 175 

Metals,  Properties  of 

Aluminum 177 

Antimony 177 

Bismuth 178 

Cadmium 178 

Copper 178 

Gold 178 

Iridium 178 

Iron 178 

Lead 178 

Magnesium 179 

Manganese 179 

Mercury : 179 

Nickel 179 

Platinum 179 

Silver 179 

Tin 179 

Zinc 179 

Miscellaneous  Materials. 

Order  of  Malleability,  etc.,  of  Metals 180 

Measures  and  Weights  of  Various  Materials 180 


CONTENTS.  XI 

PAGE 

Formulae  and  Table  for  Weight  of  Rods,  Plates,  etc 181 

Commercial  Sizes  of  Iron  and  Steel  Bars 182 

Weights  of  Iron  and  Steel  Sheets 183 

of  Iron  Bars 184 

of  Round  Steel  Bars 185 

of  Fillets 185 

of  Round,  Square,  and  Hexagon  Steel 186 

of  Plate  Iron 187 

of  Flat  Rolled  Iron 188 

of  Steel  Blooms 190 

of  Roofing  Materials 191-196 

Snow  and  Wind  Loads  on  Roofs 191 

Roof  Construction 191 

Specifications  for  Tin  and  Terne  Plates 194 

Corrugated  Sheets 194 

Weights  and  Thickness  of  Cast-iron  Pipe 196-199 

Weights  of  Cast-iron  Pipe  Columns 200 

Weight  of  Open-end  Cast-iron  Cylinders 200 

Standard  Sizes  of  Welded  Pipe 201-205 

Weight  and  Bursting  Strength  of  Welded  Pipe 205 

Tubular  Electric  Line  Poles 206 

Protective  Coatings  for  Pipes 206 

Valves  and  Fittings 206-217 

Standard  Pipe  Flanges 208-212 

Forged  Steel  Flanges 211 

Standard  Hose  Couplings 218 

Wooden  Stave  Pipe.  .  .• 218 

Riveted  Hydraulic  Pipe 219 

Riveted  Iron  Pipes 220 

Spiral  Riveted  Pipe 220 

Weight  of  Steel  for  Riveted  Pipe 221 

Bent  and  Coiled  Pipes 221 

Flexibility  of  Pipe  Bends 221 

Shelby  Cold-drawn  Steel  Tubing 222 

Seamless  Brass  and  Copper  Tubes 224,  225 

Aluminum  Tubing 226 

Lead  and  Tin-lined  Lead  Pipe 226 

Iron  Pipe  Lined  with  Tin,  Lead,  Brass,  and  Copper 227 

Weight  of  Sheet  and  Bar  Brass 228 

of  Sheet  Zinc 228 

of  Copper  and  Brass  Wire  and  Plates 229 

of  Aluminum  Sheets,  Bars,  and  Plates 230 

of  Copper  Rods 230 

Screw-threads,  U.  S.  Standard 231 

Whitworth  Screw-threads 232 

Limit-gages  for  Screw-threads .  .  .  .  : 232 

Automobile  Screws  and  Nuts 233 

International  Screw-thread 233 

Acme  Screw-thread 234 

Machine  Screws,  A.  S.  M.  E.  Standard 234 

Standard  Taps 235 

Wood  Screws 236 

Machine  Screw  Heads 237 

Set  Screws  and  Cap  Screws 238 

Weights  of  Rivets 238,  239 

Shearing  Value  of  Rivets.  Bearing  Value  of  Riveted  Plates 240 

Length  of  Rivets  for  Various  Grips 241 

Lag  Screws 241 

Weight  of  Bolts  with  Square  Heads  and  Nuts 242 

Washers .242,  243 

Hanger  Bolts 243 

Turnbuckles 243 

Track  Bolts 244 

Cut  Nails 244 

Material  Required  per  Mile  of  Railroad  Track 245 

Wire  Nails 246 

Spikes.  .,,,,,,,*,, 248 


Xii  CONTENTS. 

PAGE 

Wires  of  Different  Metals 248 

Steel  Wire,  Size,  Strength,  etc 249 

Piano  Wire 250 

Telegraph  Wire 250-252 

Plow-steel  Wire 250,  258 

Galvanized  Iron  Wire 250 

Copper  Wire,  Bare  and  Insulated 251,  252 

Notes  on  Wire  Rope 253 

Wire  Rope  Tables 255-262 

Varieties  and  Uses  of  Wire  Rope 256 

Splicing  of  Wire  Ropes 263 

Chains  and  Chain  Cables 264 

Sizes  of  Fire  Brick 266 

Refractoriness  of  American  Fire-brick 268 

Slag  Bricks  and  Slag  Blocks 268 

Magnesia  Bricks 269 

Fire  Clay  Analysis 269 

Zirconia 270 

Asbestos 270 

Standard  Cross-sections  of  Materials,  for  Draftsmen 271 

Strength  of  Materials. 

Stress  and  Strain 272 

Elastic  Limit 273 

Yield  Point 273 

Modulus  of  Elasticity 274 

Resilience 274 

Elastic  Limit  and  Ultimate  Stress 275 

Repeated  Stresses 275 

Repeated  Shocks 276 

Stresses  due  to  Sudden  Shocks 278 

Tensile  Strength 278 

Measurement  of  Elongation 279 

Shapes  of  Test  Specimens 280 

Increasing  Tensile  Strength  of  Bars  by  Twisting 280 

Compressive  Strength 281 

Columns,  Pillars,  or  Struts 283 

Hodgkinson's  Formula.     Euler's  Formula 

Gordon's  Formula.     Rankine's  Formula 

Wrought-iron  Columns 2S5 

Built  Columns 285-286 

The  Straight-line  Formula 285 

Comparison  of  Column  Formulae 286 

Tests  of  Large  Built  Steel  Columns 287 

Working  Strains  in  Bridge  Members 287 

Strength  of  Cast-iron  Columns 

Safe  Load  on  Cast-iron  Columns 291 

Strength  of  Brackets  on  Cast-iron  Columns 292 

Moment  of  Inertia 293 

Radius  of  Gyration 293 

Elements  of  Usual  Sections 

Eccentric  Loading  of  Columns 296 

Transverse  Strength 297 

Formulae  for  Flexure  of  Beams 297 

Safe  Loads  on  Steel  Beams 298,  309 

Beams  9f  Uniform  Strength 301 

Dimensions  and  Weights  of  Structural  Steel  Sections 302 

Allowable  Tension  in  Steel  Bars 305 

Properties  of  Rolled  Structural  Shapes 305 

"  Steel  I-Beams , 307 

"  Steel  Wrought  Plates 308 

"  Corrugated  Plates 310 

Spacing  of  Steel  I-Beams 311 

Properties  of  Steel  Channels 312 

"  T  Shapes 313 


CONTENTS.  .  Xlll 

PAGE 

Properties  of  Angles 316 

"  Z-bars 317 

Rivet  Spacing  for  Structural  Work 321 

Dimensions  and  Safe  Load  on  Built  Steel  Columns 323-330 

Bethlehem  Girder  and  I-beams  and  H-columns 331 

Torsional  Strength 334 

Elastic  Resistance  to  Torsion 334 

Combined  Stresses 335 

Stress  due  toTemperature 335 

Strength  of  Flat  Plates 336 

Thickness  of  Flat  Cast-iron  Plates 336 

Strength  of  Unstayed  Flat  Surfaces 337 

Unbraced  Heads  of  Boilers 337 

Strength  of  Stayed  Surfaces 338 

Stresses  in  Steel  Plating  under  Water  Pressure 338 

Spherical  Shells  and  Domed  Heads 339 

Thick  Hollow  Cylinders  under  Tension 339 

Thin  Cylinders  under  Tension 340 

Carrying  Capacity  of  Steel  Rollers  and  Balls 340 

Resistance  of  Hollow  Cylinders  to  Collapse 341,  343 

Formula  for  Corrugated  Furnaces 342 

Hollow  Copper  Balls 345 

Holding.  Power  of  Nails,  Spikes,  Bolts,  and  Screws 346 

Cut  versus  Wire  Nails 347 

Strength  of  Bolts 347 

Initial  Strain  on  Bolts 347 

Strength  of  Chains 348 

Stand  Pipes  and  their  Design 349 

Riveted  Steel  Water-pipes 351 

Kirkaldy's  Tests  of  Materials 352-358 

Cast  Iron 352 

Iron  Castings 352 

Iron  Bars,  Forgings,  etc 352--. 

Steel  Rails  and  Tires 353 

Spring  Steel,  Steel  Axles,  Shafts 354 

Riveted  Joints,  Welds 355 

Copper,  Brass,  Bronze,  etc 356 

Wire-rope 356 

Wire 357 

Ropes,  Hemp,  and  Cotton . .  357 

Belting,  Canvas 357 

Stones 357 

Brick,  Cement,  Wood 358 

Tensile  Strength  of  Wire 358 

Watertown  Testing-machine  Tests 359 

Riveted  Joints 359 

Wrought-iron  Bars,  Compression  Tests 359 

Steel  Eye-bars 360 

Wrought-iron  Columns 360 

Cold  Drawn  Steel 361 

Tests  of  Steel  Angles x .  .  362 

Shearing  Strength 362 

Relation  of  Shearing  to  Tensile  Strength 362 

Strength  of  Iron  and  Steel  Pipe 363 

Threading  Tests  of  Pipe 363 

Old  Tubes  used  as  Columns 363 

Methods  of  Testing  Hardness  of  Metals 364 

Holding  Power  of  Boiler-tubes 364 

Strength  of  Glass 365 

Strength  of  Ice 366 

Strength  of  Timber 366 

Expansion  of  Timber 367,  369 

Tests  of  American  Woods . 367 

Shearing  Strength  of  Woods 367 

Copper  at  High  Temperatures 368 

Drying  of  Wood 368 

Preservation  of  Timber « 368 


XIV  CONTENTS. 

PAGE 

Copper  Castings  of  High  Conductivity 368 

Tensile  Strength  of  Rolled  Zinc  Plates 369 

Strength  of  Brick,  Stone,  etc.  .  '. 369 

"  Lime  and  Cement  Mortar. 372 

"  Flagging .'.'.'  373 

Tests  of  Portland  Cement 373 

Moduli  of  Elasticity  of  Various  Materials 374 

Factors  of  Safety 374 

Properties  of  Cork 377 

Vulcanized  India-Rubber 378 

Specifications  for  Air  Hose 379 

Nickel 379 

Aliuninum,  Properties  and  Uses 380 

Alloys. 

Alloys  of  Copper  and  Tin,  Bronze 384 

Alloys  of  Copper  and  Zinc,  Brass 386 

Variation  in  Strength  of  Bronze 386 

Copper-tin-zinc  Alloys 387 

Liquation  or  Separation  of  Metals 388 

Alloys  used  in  Brass  Foundries 390 

Tobin  Bronze 392 

Qualities  of  Miscellaneous  Alloys 392 

Copper-zinc-iron  Alloys 393 

'  Alloys  of  Copper,  Tin,  and  Lead 394 

Phosphor  Bronze 394 

Alloys  for  Casting  under  Pressure 395 

Aluminum  Alloys 396 

Caution  as  to  Strength  of  Alloys 398 

Alloys  of  Aluminum,  Silicon,  and  Iron 398 

Tungsten-aluminum  Alloys 399 

The  Thermit  Process 400 

Aluminum-tin  Alloys 400 

Manganese  Alloys 401 

Manganese  Bronze 401 

German  Silver 402 

Monel  Metal 403 

Copper-nickel  Alloys 403 

Alloys  of  Bismuth .404 

Fusible  Alloys 404 

Bearing  Metal  Alloys 405 

Bearing  Metal  Practice,  1907 407 

White  Metal  for  Engine  Bearings 407 

Alloys  containing  Antimony 407 

White-metal  Alloys 407 

Babbitt  Metals 407,  408 

Type-metal 408 

Solders 409 

Ropes  and  Cables. 

Strength  of  Hemp,  Iron,  and  Steel  Ropes 410 

Rope  for  Hoisting  or  Transmission 411 

Cordage,  Technical  Terms  of 411 

Splicing  of  Ropes 412 

Cargo  Hoisting 414 

Working  Loads  for  Manila  Rope  .  .                     414 

Knots 415 

Life  of  Hoisting  and  Transmission  Rope 415 

Efficiency  of  Rope  Tackles 415 

Springs. 

Laminated  Steel  Springs 417 

Helical  Steel  Springs 418 


CONTENTS.  XV 

PAGE 

Carrying  Capacity  of  Springs 419 

Elliptical  Springs 423 

Springs  to  Resist  Torsional  Force 423 

Phosphor-bronze  Springs 424 

Chromium-Vanadium  Spring  Steel 424 

Test  of  a  Vanadium  Steel  Spring 424 

Riveted  Joints. 

Fairbairn's  Experiments 424 

Loss  of  Strength  by  Punching 424 

Strength  of  Perforated  Plates 424 

Hand  versus  Hydraulic  Riveting 424 

Formulae  for  Pitch  of  Rivets 427,  434 

Proportions  of  Joints 427 

Efficiencies  of  Joints 428 

Diameter  of  Rivets 429 

Shearing  Resistance  of  Rivet  Iron  and  Steel 430 

Strength  of  Riveted  Joints 431 

Riveting  Pressures 435 

Tests  of  Soft  Steel  Rivets 435 

Iron  and  Steel. 

Classification  of  Iron  and  Steel 436 

Grading  of  Pig  Iron 437 

Manufacture  of  Cast  Iron 437 

Influence  of  Silicon  Sulphur,  Phos.  and  Mn  on  Cast  Iron 438 

Microscopic  Constituents • 439 

Analyses  of  Cast  Iron 439 

Specifications  for  Pig  Iron  and  Castings 441,  443 

Specifications  for  Cast-iron  Pipe 441 

Chemical  Standards  for  Castings 441 

Strength  of  Cast  Iron 444,  451 

Strength  in  Relation  to  Cross-section 446,  447 

"  Semi-steel " 446,  453 

Shrinkage  of  Cast  Iron 447 

White  Iron  Converted  into  Gray 448 

Mobility  of  Molecules  of  Cast  Iron 449 

Expansion  of  Iron  by  Heat 449,  465 

Permanent  Expansion  of  Cast  Iron  by  Heating 449 

Castings  from  Blast  Furnace  Metal 450 

Effect  of  Cupola  Melting 450 

Additions  of  Titanium,  etc.,  to  Cast  Iron 450,  451 

Mixture  of  Cast  Iron  with  Steel 453 

Bessemerized  Cast  Iron 453 

Bad  Cast  Iron «-. 453 

Malleable  Cast  Iron ! 454 

Design  of  Malleable  Castings 457 

Specifications  of  Malleable  Iron 457 

Strength  of  Malleable  Cast  Iron 458 

Wrought  Iron 459 

Chemistry  of  Wrought  Iron 460 

Electrolytic  Iron 460 

Influence  of  Rolling  on  Wrought  Iron 460 

Specifications  for  Wrought  Iron 461 

Stay-bolt  Iron 462 

Tenacity  of  Iron  at  High  Temperatures 463 

Effect  of  Cold  on  Strength  of  Iron 464 

Durability  of  Cast  Iron 465 

Corrosion  of  Iron  and  Steel 466 

Corrosion  of  Iron  and  Steel  Pipes 467 

Electrolytic  Theory,  and  Prevention  of  Corrosion 468 

Chrome  Paints,  Anti-corrosive 469 

Corrosion  Caused  by  Stray  Electric  Currents 470 

Electrolytic  Corrosion  due  to  Overstrain 470 


XVI  CONTENTS. 

PAGE 

Preservative  Coatings,  Paints,  etc 471 

Inoxydation  Processes,  Bower-Barff,  etc 472 

Aluminum  Coatings 473 

Galvanizing 473 

Sherardizing,  Galvanizing  by  Cementation 474 

Lead  Coatings 474 

Steel. 

Manufacture  of  Steel 475 

Crucible,  Bessemer,  and  Open  Hearth  Steel 475 

Relation  between  Chemical  and  Physical  Properties 476 

Electric  Conductivity 477 

"  Armco  Ingot  Iron  " 477 

Variation  in  Strength 477,  478 

Bending  Tests  of  Steel 478 

Effect  of  Heat  Treatment  and  of  Work 478 

Hardening  Soft  Steel 479 

Effect  of  Cold  Rolling 479 

Comparison  of  Full-sized  and  Small  Pieces 480 

Recalescence  of  Steel 480 

Critical  Point 480 

Metallography 480 

Burning,  Overheating,  and  Restoring  Steel 481 

Working  Steel  at  a  Blue  Heat 482 

Oil  Tempering  and  Annealing 482 

Brittleness  due  to  Long-continued  Heating 483 

Influence  of  Annealing  upon  Magnetic  Capacity 483 

Treatment  of  Structural  Steel 483 

May  Carbon  be  Burned  out  of  Steel? 485 

Effect  of  Nicking  a  Bar 485 

Dangerous  Low  Carbon  Steel 486 

Specific  Gravity 486 

Occasional  Failures 486 

Segregation  in  Ingots  and  Plates 487 

Endurance  of  Steel  under  Repeated  Stresses 487 

Welding  of  Steel 488 

The  Thermit  Welding  Process 488 

Oxy-acetylene  Welding  and  Cutting  of  Metals 488 

Hydraulic  Forging 488 

Fluid-compressed  .Steel 488 

Steel  Castings 489 

Crucible  Steel 490 

Effect  of  Heat  on  Gram 491 

Heating  and  Forging 491 

Tempering  Steel 493 

Kinds  of  Steel  used  for  Different  Purposes 494 

High-speed  Tool  Steel 

Manganese  Steel- 494 

Chrome  Steel 496 

Aluminum  Steel 496 

Tungsten  Steel 496 

Nickel  Steel 497 

Copper  Steel 499 

Nickel- Vanadium  Steel 499 

Static  and  Dynamic  Properties  of  Steel 500 

Strength  and  Fatigue  Resistance  of  Steels 501 

Chromium- Vanadium  Steel • 502 

Heat  Treatment  of  Alloy  Steels 502,  503 

Specifications  for  Steel 504-51 1 

High-strength  Steel  for  Shipbuilding 507 

Fire-box  Steel 508 

Steel  Rails 508 

MECHANICS. 

Matter,  Weight,  Mass 511 

Force,  Unit  of  Force 512 


CONTENTS.  XVli 

PAGE 

Local  Weight * 512 

Inertia 513 

Newton's  Laws  of  Motion 513 

Resolution  of  Forces 513 

Parallelogram  of  Forces 513 

Moment  of  a  Force .      .  514 

Statical  Moment,  Stability 515 

Stability  of  a  Dam 515 

Parallel  Forces 515 

Couples 515 

Equilibrium  of  Forces 516 

Center  of  Gravity 516 

Moment  of  Inertia 517 

Centers  of  Oscillation  and  Percussion 518 

Center  and  Radius  of  Gyration 518 

The  Pendulum 520 

Conical  Pendulum 520 

Centrifugal  Force 521 

Velocity,  Acceleration,  Falling  Bodies 521 

Value  of  g 522 

Angular  Velocity 522 

Height  due  to  Velocity 523 

Parallelogram  of  Velocities 522 

Velocity  due  to  Falling  a  Given  Height 524 

Fundamental  Equations  in  Dynamics 525 

Force  of  Acceleration 526 

Formulae  for  Accelerated  Motion 527 

Motion  on  Inclined  Planes '.- 527 

Momentum 527 

Work,  Energy,  Power 528 

Work  of  Acceleration 529 

Work  of  Accelerated  Rotation .  . 529 

Force  of  a  Blow 529 

Impact  of  Bodies 530 

Energy  of  Recoil  of  Guns 531 

Conservation  of  Energy.  .  . T 531 

Sources  of  Energy 531 

Perpetual  Motion 532 

Efficiency  of  a  Machine 532 

Animal-power,  Man-power 532 

Man-wheel,  Tread  Mills 533 

Work  of  a  Horse 533 

Horse-gin 534 

Resistance  of  Vehicles 534 


Elements  of  Mechanics. 

The  Lever.  .  535 

The  Bent  Lever 536 

The  Moving  Strut 536 

The  Toggle-joint 536 

The  Inclined  Plane 537 

The  Wedge 537 

The  Screw 537 

The  Cam 537 

Efficiency  of  a  Screw 538 

Efficiency  of  Screw  Bolts 538 

Pulleys  or  Blocks '. 539 

Differential  Pulley 539 

Wheel  and  Axle 539 

Toothed- wheel  Gearing 539 

Endless  Screw,  Worm  Gear 540 

Differential  Windlass 540 

Differential  Screw 540 

Efficiency  of  a  Differential  Screw 641 


XV1U  CONTENTS. 

Stresses  in  Framed  Structures. 

Cranes  and  Derricks 541 

Shear  Poles  and  Guys 542 

King  Post  Truss  or  Bridge 543 

Queen  Post  Truss 543 

Burr  Truss 544 

Pratt  or  Whipple  Truss 544 

Method  of  Moments 545 

Howe  Truss 546 

Warren  Girder 546 

Roof  Truss 547 

The  Economical  Angle 548 

HEAT. 

Thermometers  and  Pyrometers 549 

Centigrade  and  Fahrenheit  degrees  compared . 550 

Temperature  Conversion  Table 552 

Copper-ball  Pyrometer 553 

Thermo-electric  Pyrometer 554 

Temperatures  in  Furnaces 554 

Seger's  Fire-clay  Pyrometer 555 

Wiborgh  Air  Pyrometer 655 

Mesure  and  Nouel's  Pyrometer 556 

Uehling  and  Steinbart  Pyrometer 557 

Air- thermometer 557 

High  Temperatures  Judged  by  Color 558 

Boiling-points  of  Substances 559 

Melting-points 559 

Unit  of  Heat 560 

Mechanical  Equivalent  of  Heat 560 

Heat  of  Combustion 560 

Heat  Absorbed  by  Decomposition 561 

Specific  Heat 562 

Thermal  Capacity  of  Gases 564 

Expansion  by  Heat 565 

Absolute  Temperature,  Absolute  Zero 567 

Latent  Heat  of  Fusion §68 

Latent  Heat  of  Evaporation 568 

Total  Heat  of  Evaporation 569 

Evaporation  and  Drying 569 

Evaporation  from  Reservoirs • 569 

Evaporation  by  the  Multiple  System 570 

Resistance  to  Boiling *  . .  570 

Manufacture  of  Salt 570 

Solubility  of  Salt 571 

Salt  Contents  of  Brines 571 

Concentration  of  Sugar  Solutions 572 

Evaporating  by  Exhaust  Steam 572 

Drying  in  Vacuum 573 

Driers  and  Drying 574 

Design  of  Drying  Apparatus 576 

Humidity  Table 577 

Radiation  of  Heat 578 

Black-body  Radiation 579 

Conduction  and  Convection  of  Heat 579 

Rate  of  External  Conduction 580 

Heat  Conduction  of  Insulating  Materials 581 

Heat  Resistance,  Reciprocal  of  Heat  Conductivity 582 

Steam-pipe  Coverings 584 

Transmission  through  Plates 587 

Transmission  in  Condenser  Tubes 588 

Transmission  of  Heat  in  Feed-water  Heaters 590 

Transmission  through  Cast-iron  Plates 591 

Heating  Water  by  Steam  Coils 591 

Transmission  from  Air  or  Gases  to  Water 592 


CONTENTS.  XIX 

PAGE 

Transmission  from  Flame  to  Water 593 

Cooling  of  Air 594 

Transmission  from  Steam  or  Hot  Water  to  Air 595 

Thermodynamics 597 

Entropy 599 

Reversed  Carnot  Cycle,  Refrigeration .  .  , 600 

Principal  Equations  of  a  Perfect  Gas 600 

Construction  of  the  Curve  PV«  =  C 602 

Temperature-Entropy  Diagram  of  Water  and  Steam 602 

PHYSICAL  PROPERTIES   OF   GASES. 

Expansion  of  Gases 603 

Boyle  and  Marriotte's  Law 603 

Law  of  Charles,  Avogadro's  Law 604 

Saturation  Point  of  Vapors 604 

Law  of  Gaseous  Pressure 604 

Flow  of  Gases 605 

Absorption  by  Liquids 605 

Liquefaction  of  Gases,  Liquid  Air 605 

AIR. 

Properties  of  Air 606 

Barometric  Pressures 606 

Air-manometer 607 

Conversion  Table  for  Air  Pressures 607 

Pressure  at  Different  Altitudes 607,  609 

Leveling  by  the  Barometer  and  by  Boiling  Water 607 

To  find  Difference  in  Altitude 608 

Weight  of  Air  at  Different  Pressures  and  Temperatures 609 

Moisture  in  Atmosphere 609,  611 

Humidity  Table 610 

Weight  of  Air  and  Mixtures  of  Air  and  Vapor 610,  613 

Specific  Heat  of  Air 614 

Flow  of  Air. 

Flow  of  Air  through  Orifices 615 

Flow  of  Air  in  Pipes 617 

Tables  of  Flow  of  Air 622,  623 

Effects  of  Bends  in  Pipe 624 

Anemometer  Measurements 624 

Equalization  of  Pipes 625 

Wind. 

Force  of  the  Wind 626 

Wind  Pressure  in  Storms 627 

Windmills 627 

Capacity  of  Windmills 629 

Economy  of  Windmills 630 

Electric  Power  from  Windmills 632 

Compressed  Air. 

Heating  of  Air  by  Compression 632 

Loss  of  Energy  in  Compressed  Air 632 

Loss  due  to  Heating 633 

Work  of  Adiabatic  Compression  of  Air 634 

Compound  Air-compression 635 


XX  CONTENTS. 

PAGE 

Mean  Effective  Pressures 635,  636 

Horse-power  Required  for  Compression 637 

Compressed-air  Engines 638 

Mean  and  Terminal  Pressures 638 

Air-compression  at  Altitudes 639 

Popp  Compressed-air  System 639 

Small  Compressed-air  Motors 640 

Efficiency  of  Air-heating  Stoves 640 

Efficiency  of  Compressed-air  Transmission 640 

Efficiency  of  Compressed-air  Engines 640 

Air-compressors .- 641 

Tests  of  Air  compressors 643 

Steam  Required  to  Compress  100  Cu.  Ft.  of  Air. 644 

Requirements  of  Rock-drills 645 

Compressed  Air  for  Pumping  Plants 645 

Compressed  Air  for  Hoisting  Engines 646 

Practical  Results  with  Air  Transmission 647 

Effect  of  Intake  Temperature 647 

Compressed-air  Motors  with  Return  Circuit 648 

Intercoolers  for  Air-compressors 64.8 

Centrifugal  Air-compressors 648 

High-pressure  Centrifugal  Fans 649 

Test  of  a  Hydraulic  Air-compressor 650 

Mekarski  Compressed-air  Tramways 652 

Compressed  Air  Working  Pumps  in  Mines .  , 652 

Compressed  Air  for  Street  Railways 652 

Fans  and  Blowers. 

Centrifugal  Fans 653 

Best  Proportions  of  Fans 653 

Pressure  due  to  Velocity 653 

Blast  Area  or  Capacity  Area 655 

Pressure  Characteristics  of  Fans 655 

Quantity  of  Air  Delivered 655 

Efficiency  of  Fans  and  Positive  Blowers 657 

Tables  of  Centrifugal  Fans t 658-666 

Effect  of  Resistance  on  Capacity  of  Fans 664 

Sirocco  or  Multivane  Fans '664 

Methods  of  Testing  Fans 667 

Horse-power  of  a  Fan 668 

Pitot  Tube  Measurements 669 

Thomas  Electric  Air  and  Gas  Meter 669 

Flow  of  Air  through  an  Orifice 670 

Diameter  of  Blast-pipes 670 

Centrifugal  Ventilators  for  Mines 672 

Experiments  on  Mine  Ventilators 673 

Disk  Fans » 675 

Efficiency  of  Disk  Fans 676 

Positive  Rotary  Blowers 677 

Steam-jet  Blowers  and  Exhausters 679 

Blowing  Engines 680 

HEATING  AND  VENTILATION. 

Ventilation 681 

Quantity  of  Air  Discharged  through  a  Ventilating  Duct 683 

Heating  and  Ventilating  of  Large  Buildings 684 

Comfortable  Temperatures  and  Humidities 685 

Carbon  Dioxide  Allowable  in  Factories 685 

Standards  of  Ventilation 686 

Air  Washing 687 

Contamination  of  Air 687 

Standards  for  Calculating  Heating  Problems 687 


CONTENTS.  XXI 

PAGE 

Heating  Value  of  Coal 687 

Heat  Transmission  through  Walls,  etc 688 

Allowance  for  Exposure  and  Leakage 689 

Heating  by  Hot-air  Furnaces .  690 

Carrying  Capacity  of  Air-pipes 691 

Volume  of  Air  at  Different  Temperatures 692 

Sizes  of  Pipes  Used  in  Furnace  Heating 692 

Furnace  Heating  with  Forced  Air  Supply 693 

Rated  Capacity  of  Boilers  for  House  Heating 693 

Capacity  of  Grate-surface 694 

Steam  Heating,  Rating  of  Boilers 694 

Testing  Cast-iron  Heating  Boilers 696 

Proportioning  House  Heating  Boilers 696 

Coefficient  of  Transmission  in  Direct  Radiation 697 

Heat  Transmitted  in  Indirect  Radiation 698 

Short  Rules  for  Computing  Radiating  Surface 698 

Carrying  Capacity  of  Steam  Pipes  in  Low  Pressure  .Heating ....  698 

Proportioning  Pipes  to  Radiating  Surface 700 

Sizes  of  Pipes  in  Steam  Heating  Plants 701 

Resistance  of  Fittings 701 

Removal  of  Air,  Vacuum  Systems 702 

Overhead  Steam-pipes 702 

Steam-consumption  in  Car-heating 702 

Heating  a  Greenhouse  by  Steam 702 

Heating  a  Greenhouse  by  Hot  Water 703 

Hot-water  Heating 703 

Velocity  of  Flow  in  Hot- water  Heating 703 

Sizes  of  Pipe  for  Hot- water  Heating 704 

Sizes  of  Flow  and  Return  Pipes 705 

Heating  by  Hot-water,  with  Forced  Circulation 707 

Corrosion  of  Pipe  in  Hot- water  Heating 708 

Blower  System  of  Heating  and  Ventilating 708 

Advantages  and  Disadvantages  of  the  Plenum  System 708 

Heat  Radiated  from  Coils  in  the  Blower  System 708 

Test  of  Cast-iron  Heaters  for  Hot-blast  Work 709 

Factory  Heating  by  the  Fan  System 710 

Artificial  Cooling  of  Air 710 

Capacities  of  Fans  for  Hot-blast  Heating 711 

Relative  Efficiency  of  Fans  and  Heated  Chimneys 712 

Heating  a  Building  to  70°  F 712 

Heating  by  Electricity 713 

Mine- ventilation 714 

Friction  of  Air  in  Underground  Passages 714 

Equivalent  Orifices - 715 

WATER. 

Expansion  of  Water 716 

Weight  of  Water  at  Different  Temperatures. 716,  717 

Pressure  of  Water  due  to  its  Weight 718,  719 

Head  Corresponding  to  Pressures 718 

Buoyancy 719 

Boiling-point 719 

Freezing-point 719 

Sea-water 719 

Ice  and  Snow 720 

Specific  Heat  of  Water 720 

Compressibility  of  Water '. . .  720 

Impurities  of  Water 720 

Causes  of  Incrustation 721 

Means  for  Preventing  Incrustation 721 

Analyses  of  Boiler-scale 722 

Hardness  of  Water 723 

Purifying  Feed-water 723 

Softening  Hard  Water 724 


XX11  CONTENTS. 

Hydraulics.    Flow  of  Water.  PAGE 

Formulae  for  Discharge  through  Orifices  and  Weirs 726 

Flow  of  Water  from  Orifices 727 

Flow  in  Open  and  Closed  Channels 728 

General  Formulae  for  Flow .  .  .  : 728 

Chezy's  Formula 728 

Values  of  the  Coefficient  c 728,  732 

Table,  Fall  in_Feet  per  mile,  etc 729 

Values  of  \/r  for  Circular  Pipes 730 

Kutter's  Formula 730 

D'Arcy's  Formula 732 

Values  of  a  \/r  for  Chezy's  Formula 733 

Values  of  the  Coefficient  of  Friction 734 

Loss  of  Head 735 

Resistance  at  the  Inlet  of  a  pipe 735 

Exponential  Formulae,  Williams'  and  Hazen's  Tables 736 

Short  Formulas 737 

Flow  of  Water  in  a  20-inch  Pipe , 737 

Coefficients  for  Reducing  H.  and  W.  to  Chezy's  Formula 737 

Tables  of  Flow  of  Water  in  Circular  Pipes 738-743 

Flow  of  Water  in  Riveted  Pipes 743 

Long  Pipe  Lines 743 

Flow  of  Water  in  House-service  Pipes 744 

Friction  Loss  in  Clean  Cast-iron  Pipe 745 

Approximate  Hydraulic  Formulae 746 

Compound  Pipes,  and  Pipes  with  Branches 746 

Rifled  Pipes  for  Conveying  Oils 746 

Effect  of  Bend  and  Curves 747 

Loss  of  Pressure  Caused  by  Valves,  etc 747,  748 

Hydraulic  Grade-line 748 

Air-bound  Pipes 748 

Water  Hammer 749 

Vertical  Jets 749 

Water  Delivered  through  Meters 749 

Price  Charged  for  Water  in  Cities 749 

Fire  Streams 749 

Hydrant  Pressures  Required  with  Different  Lengths  and  Sizes  of 

Hose 750 

Pump  Inspection  Table 751 

Pipe  Sizes  for  Ordinary  Fire  Streams 752 

Friction  Losses  in  Hose 752 

Rated  Capacity  of  Steam  Fire-engines 752 

Flow  of  Water  through  Nozzles 753 

The  Siphon 754 

Velocity  of  Water  in  Open  Channels 755 

Mean  Surface  and  Bottom  Velocities 755 

Safe  Bottom  and  Mean  Velocities 755 

Resistance  of  Soil  to  Erosion 755 

Abrading  and  Transporting  Power  of  Water 755 

Frictional  Resistance  of  Surfaces  Moved  in  Water 756 

Grade  of  Sewers 757 

Measurement  of  Flowing  Water 757 

Piezometer 757 

Pitot  Tube  Gauge 

Maximum  and  Mean  Velocities  in  Pipes. 758 

The  Venturi  Meter 758 

Measurement  of  Discharge  by  Means  of  Nozzles 759 

The  Lea  V-notch  Recording  Meter 759 

Flow  through  Rectangular  Orifices 760 

Measurement  of  an  Open  Stream 760 

Miners'  Inch  Measurements 761 

Flow  of  Water  over  Weirs 762 

Francis's  Formica  for  Weirs 762 

Weir  Table 763 

Bazin's  Experiments 763 

The  Cippoleti,  or  Trapezoidal  Weir 764 

The  Triangular  Weir :...... 764 


CONTENTS.  xxiii 

WATER-POWER. 

Power  of  a  Fall  of  Water 765 

Horse-power  of  a  Running  Stream 765 

Current  Motors 765 

Bernouilli's  Theorem 765 

Maximum  Efficiency  of  a  Long  Conduit 766 

Mill-power .     766 

Value  of  Water-power 76(j 

Water  Wheels.    Hydraulic  Turbines. 

Theory  of  Turbines 768 

Determination  of  Dimensions  of  Turbine  Runners 769A 

Comparison  of  Formulae  for  Dimensions  of  Turbines 769A 

Comparison  of  American  High  Speed  Runners 770 

Type  Characteristics  of  Turbines 770 

Specific  Discharge 770B 

Use  of  Type  Characteristics  to  Determine  Size  and  Type  of 

Turbines 770B 

Classes  of  Radial  Inward  Flow  Turbines 771 

Estimating  Weight  of  Turbines 771A 

Selection  of  Turbines 771A 

Eifficiency  of  Turbine  wheels 771s 

Relation  of  Efficiency  and  Water  Consumption  to  Speed ......  772 

Tests  at  the  Philadelphia  Exposition 772 

Relation  of  Gare  Openings  to  Efficiency 773 

Tests  of  Turbine  Discharge  by  Salt  Solution 774' 

Efficiency  Tables  for  Turbines 776-777 

Draft  Tubes 778 

Recent  Turbine  Practice 778 

Some  Large  Turbines 779 

The  Fall-increaser  for  Turbines 780 

Tangential  or  Impulse  Water  Wheels.- 

The  Pelton  Water  Wheel 780 

Considerations  in  the  Choice  of  a  Tangential  Wheel 781 

Control  of  Tangential  Water  Wheels 781 

Efficiency  of  the  Doble  Nozzle 782 

Tests  of  a  12-inch  Doble  Motor 782 

Water-power  Plants  Operating  under  High  Pressures 782 

Amount  of  Water  Required  to  Develop  a  Given  Horse-Power .  783 

Formulae  for  Calculating  the  Power  of  Jet  Water  Wheels 784 

Tangential  Water-wheel  Table 787 

The  Power  of  Ocean  Waves. 

Energy  of  Deep  Sea  Waves 786 

Utilization  of  Tidal  Power 787 

PUMPS    AND    PUMPING    ENGINES. 

Theoretical  Capacity  of  a  Pump 788 

Depth  of  Suction 788 

The  Deane  Pump 7X9 

Sizes  of  Direct-acting  Pumps 789,  791 

Amount  of  Water  Raised  by  a  Single-acting  Lift-pump 790 

Proportioning  the  Steam-cylinder  of  a  Direct-acting  Pump 790 

Speed  of  Water  through  Pipes  and  Pump-passages 790 

Efficiency  of  Small  Pumps 790 

The  Worthington  Duplex  Pump 791 

Speed  of  Piston 791-792 

Speed  of  Water  through  Valves 792 

Underwriters'  Pumps,  Standard  Sizes 792 

Boiler-feed  Pumps 792 

Pump  Valves 793 

The  Worthington  High-duty  Pumping  PJngine 793 


CONTENTS. 

The  d'Auria  Pumping  Engine 793 

A  72,000,000-Gallon  Pumping  Engine 793 

The  Screw  Pumping  Engine 794 

Finance  of  Pumping  Engine  Economy 794 

Cost  of  Pumping  1000  Gallons  per  Minute 795 

Centrifugal  Pumps 796 

Design  of  a  Four-stage  Turbine  Pump 797 

Relation  of  Peripheral  Speed  to  Head 797 

Tests  of  De  Laval  Centrifugal  Pump 798 

A  High-duty  Centrifugal  Pump 801 

Rotary  Pumps 801 

Tests  of  Centrifugal  and  Rotary  Pumps 802 

Duty  Trials  of  Pumping  Engines 802 

Leakage  Tests  of  Pumps 803 

Notable  High-duty  Pump  Records 805 

Vacuum  Pumps 806 

The  Pulsometer 806 

The  Jet  Pump 807 

The  Injector 807 

Pumping  by  Compressed  Air 808 

Gas-engine  Pumps ;  The  Humphrey  Gas  Pump 808 

Air-lift  Pump 808 

Air-lifts  for  Deep  Oil-wells 809 

The  Hydraulic  Ram 810 

Quantity  of  Water  Delivered  by  the  Hydraulic  Ram 810 

Hydraulic  Pressure  Transmission. 

Energy  of  Water  under  Pressure 812 

Efficiency  of  Apparatus 812 

Hydraulic  Presses 813 

Hydraulic  Power  in  London 814 

Hydraulic  Riveting  Machines 814 

Hydraulic  Forging 814 

Hydraulic  Engine 815 

FUEL. 

Theory  of  Combustion 816 

Analyses  of  the  Gases  of  Combustion 817 

Temperature  of  the  Fire 818 

Classification  of  Solid  Fuels 818 

Classification  of  Coals 819 

Analyses  of  Coals 820 

Caking  and  Non-Caking  Coals 820 

Cannel  Coals 821 

Rhode  Island  Graphitic  Anthracite 821 

Analysis  and  Heating  Value  of  Coals 821-828 

Approximate  Heating  Values 822 

Lord  and  Haas's  Tests 823 

Sizes  of  Anthracite  Coal 823 

Space  occupied  by  Anthracite 823 

Bernice  Basin,  Pa.,  Coal 824 

Connellsville  Coal  and  Coke 824 

Bituminous  Coals  of  the  Western  States 824 

Analysis  of  Foreign  Coals 825 

Sampling  Coal  for  Analyses 825 

Relative  Value  of  Steam  Coals —  826 

Calorimetric  Tests  of  Coals 826 

Classified  Lists  of  Coals 828-830 

Purchase  of  Coal  Under  Specifications 830 

Weathering  of  Coal 830 

Pressed  Fuel 831 

Spontaneous  Combustion  of  Coal 832 

Coke 832 

Experiments  in  Coking 833 

Coal  Washing 833 


CONTENTS.  XXV 

PAGE 

Recovery  of  By-products  in  Coke  Manufacture 833 

Generation  of  Steam  from  the  Waste  Heat  and  Gases  from  Coke- 
ovens 834 

Products  of  the  Distillation  of  Coal 834 

Wood  as  Fuel 835 

Heating  Value  of  Wood 835 

Composition  of  Wood 835 

Charcoal 836 

Yield  of  Charcoal  from  a  Cord  of  Wood 836 

Consumption  of  Charcoal  in  Blast  Furnaces 837 

Absorption  of  Water  and  of  Gases  by  Charcoal 837 

Miscellaneous  Solid  Fuels 837 

Dust-fuel — Dust  Explosions 837 

Peat  or  Turf 838 

Sawdust  as  Fuel 838 

Wet  Tan-bark  as  Fuel 838 

Straw  as  Fuel 839 

Bagasse  as  Fuel  in  Sugar  Manufacture 839 

Liquid  Fuel. 

Products  of  Distillation  of  Petroleum 840 

Lima  Petroleum 840 

Value  of  Petroleum  as  Fuel 840 

Fuel  Oil  Burners 842 

Specifications  for  Purchase  of  Fuel  Oil 843 

Alcohol  as  Fuel 843 

Specific  Gravity  of  Ethyl  Alcohol 844 

Vapor  Pressures  of  Saturation  of  Alcohol  and  other  Liquids ....  844 

Fuel  Gas. 

Carbon  Gas 845 

Anthracite  Gas 845 

Bituminous  Gas 846 

Water  Gas 846 

Natural  Gas  in  Ohio  and  Indiana 847 

Natural  Gas  as  a  Fuel  for  Boilers 847 

Producer-gas  from  One  Ton  of  Coal 848 

Combustion  of  Producer-gas 849 

Proportions  of  Gas  Producers  and  Scrubbers 849 

Gas  Producer  Practice 851 

Capacity  of  Producers 851 

High  Temperature  Required  for  Production  of  CO 852 

The  Mond  Gas  Producer 852 

Relative  Efficiency  of  Different  Coals  in  Gas-engine  Tests 853 

Use  of  Steam  in  Producers  and  Boiler  Furnaces 854 

Gas  Analyses  by  Volume  and  by  Weight 854 

Gas  Fuel  for  Small  Furnaces 854 

Blast-furnace  Gas 855 

Acetylene  and  Calcium  Carbide. 

Acetylene 855 

Calcium  Carbide 856 

Acetylene  Generators  and  Burners 857 

The  Acetylene  Blowpipe 857 

Ignition  Temperature  of  Gases 858 

Illuminating  Gas. 

Coal-gas 858 

Water-gas 858 

Analyses  of  Water-gas  and  Coal-gas 860 

Calorific  Equivalents  of  Constituents 860 

Efficiency  of  a  Water-gas  Plant 861 

Space  Required  for  a  Water-gas  Plant 862 

Fuel- value  of  Illuminating  Gas 863 


XXVI  CONTENTS. 

PAGE 

Flow  of  Gas  in  Pipes. **»»,» 864-866 

Services  for  Lamps 864 

Factors  for  Reducing  Volumes  of  Gas 865 

STEAM. 

Temperature  and  Pressure 867 

Total  Heat 867 

Latent  Heat  of  Steam 867 

Specific  Heat  of  Saturated -Steam 867 

The  Mechanical  Equivalent  of  Heat 868 

Pressure  of  Saturated  Steam 868 

Volume  of  Saturated  Steam 868 

Specific  Heat  of  Superheated  Steam 869 

Specific  Density  of  Gaseous  Steam 870 

Table  of  the  Properties  of  Saturated  Steam 871-874 

Table  of  the  Properties  of  Superheated  Steam 874,  875 

Flow  of  Steam. 

Flow  of  Steam  through  a  Nozzle 876 

Napier's  Approximate  Rule 876 

Flow  of  Steam  in  Pipes 877 

Flow  of  Steam  in  Long  Pipes,  Ledoux's  Formula 877 

Table  of  Flow  of  Steam  in  Pipes 878 

Carrying  Capacity  of  Extra  Heavy  Steam  Pipes 879 

Resistance  to  Flow  by  Bends,  Valves,  etc 879 

Sizes  of  Steam-pipes  for  Stationary  Engines 879 

Sizes  of  Steam-pipes  for  Marine  Engines 880 

Proportioning  Pipes  for  Minimum  Loss  by  Radiation  and  Friction  880 

Available  Maximum  Efficiency  of  Expanded  Steam 881 

Steam-pipes. 

Bursting-tests  of  Copper  Steam-pipes 882 

Failure  of  a  Copper  Steam-pipe 882 

Wire-wound  Steam-pipes 882 

Materials  for  Pipes  and  Valves  for  Superheated  Steam 882 

Riveted  Steel  Steam-pipes 883 

Valves  in  Steam-pipes 883 

The  Steam  Loop 883 

Loss  from  an  Uncovered  Steam-pipe 884 

Condensation  in  an  Underground  Pipe  Line 884 

Steam  Receivers  in  Pipe  Lines 884 

Equation  of  Pipes 884 

Identification  of  Power  House  Piping  by  Colors 885 

THE   STEAM-BOILER. 

The  Horse-power  of  a  Steam-boiler 885 

Measures  for  Comparing  the  Duty  of  Boilers 886 

Unit  of  Evaporation 886 

Steam-boiler  Proportions 887 

Heating-surface 887 

Horse-power,  Builders'  Rating 888 

Grate-surface 888 

Areas  of  Flues 889 

Air-passages  Through  Grate-bars 889 

Performance  of  Boilers 889 

Conditions  which  Secure  Economy 890 

Air  Leakage  in  Boiler  Settings 891 

Efficiency  of  a  Boiler 891 

Autographic  CO2  Recorders 891 

Relation  of  Efficiency  to  Rate  of  Driving,  Air  Supply,  etc 893 

Effect  of  Quality  of  Coal  upon  Efficiency 895 

Effect  of  Imperfect  Combustions  and  Excess  Air  Supply 896 

Theoretical  Efficiency  with  Pittsburgh  Coal 896 


CONTENTS.  XXVII 

/ 

The  Straight  Line  Formula  for  Efficiency 896 

High  Rates  of  Evaporation 898 

Boilers  Using  Waste  Gases 898 

Maximum  Efficiencies  at  Different  Rates  of  Driving 898 

Rules  for  Conducting  Boiler  Tests 899 

Heat  Balance  in  Boiler  Tests 907 

Factors  of  Evaporation 908 

Strength  of  Steam-boilers. 

Rules  for  Construction 908 

Shell-plate  Formulae 913 

Efficiency  of  Riveted  Joints 914 

Loads  Allowed  on  Stays 916 

Holding  Power  of  Boiler  Tubes 916 

Safe-working  Pressures 918 

Boiler  Attachments,  Furnaces,  etc. 

Fusible  Plugs 2 918 

Steam  Domes 918 

Mechanical  Stokers 918 

The  Hawley  Down-draught  Furnace 919 

Under-feed  Stokers 919 

Smoke  Prevention 920 

Burning  Illinois  Coal  without  Smoke 921 

Conditions  of  Smoke  Prevention 922 

Forced  Combustion 923 

Fuel  Economizers 924 

Thermal  Storage 927 

Incrustation  and  Corrosion 927 

Boiler-scale  Compounds 929 

Removal  of  Hard  Scale 930 

Corrosion  in  Marine  Boilers 930 

Use  of  Zinc '. 931 

Effect  of  Deposit  on  Flues 931 

Dangerous  Boilers 932 

Safety-valves. 

Rules  for  Area  of  Safety-valves 932 

Spring-loaded  Safety-valves 933 

Safety  Valves  for  Locomotives 935 

The  Injector. 

Equation  of  the  Injector 936 

Performance  of  Injectors 937 

Boiler-feeding  Pumps 937 

Feed-water  Heaters. 

Percentage  of  Saving  Due  to  Use  of  Heaters 938 

Strains  Caused  by  Cold  Feed-water 939 

Calculation  of  Surface  of  Heaters  and^Condensers 939 

Open  vs.  Closed  Feed-water  Heaters 940 

Steam  Separators. 

Efficiency  of  Steam  Separators 941 

Determination  of  Moisture  in  Steam. 

Steam  Calorimeters 942 

Coil  Calorimeter 942 

Throttling  Calorimeters 943 

Separating  Calorimeters 943 


XXV111  CONTENTS. 

PAGE 

Identification  of  Dry  Steam 944 

Usual  Amount  of  Moisture  in  Steam 944 

Chimneys. 

Chimney  Draught  Theory 944 

Force  of  Intensity  of  Draught 945 

Rate  of  Combustion  Due  to  Height  of  Chimney 947 

High  Chimneys  not  Necessary 948 

Height  of  Chimneys  Required  for  Different  Fuels 948 

Protection  of  Chimney  from  Lightning 949 

Table  of  Size  of  Chimneys 950 

Velocity  of  Gas  in  Chimneys 951 

Size  of  Chimneys  for  Oil  Fuel 951 

Chimneys  with  Forced  Draught 952 

Largest  Chimney  in  the  World 952 

Some  Tall  Brick  Chimneys 953,  954 

Stability  of  Chimneys 954 

Steel  Chimneys 956 

Reinforced  Concrete  Chimneys 958 

Sheet-iron  Chimneys 958 

THE  STEAM  ENGINE. 

•  Expansion  of  Steam ; 959 

Mean  and  Terminal  Absolute  Pressures 960 

Calculation  of  Mean  Effective  Pressure 961 

Mechanical  Energy  of  Steam  Expanded  Adiabatically 963 

Measures  for  Comparing  the  Duty  of  Engines 963 

Efficiency,  Thermal  Units  per  Minute 964 

Real  Ratio  of  Expansion 965 

Effect  of  Compression 965 

Clearance  in  Low-  and  High-speed  Engines 966 

Cylinder-condensation 966 

Water-consumption  of  Automatic  Cut-off  Engines 967 

Experiments  on  Cylinder-condensation 967 

Indicator  Diagrams 968 

Errors  of  Indicators 969 

Pendulum  Indicator  Rig 969 

The  Manograph 969 

The  Lea  Continuous  Recorder 970 

Indicated  Horse-power 970 

Rules  for  Estimating  Horse-power 970 

Horse-power  Constants 971 

Table  of  Engine  Constants 972 

To  Draw  Clearance  on  Indicator-diagram 974 

To  Draw  Hyperbola  Curve  on  Indicator-diagram 974 

Theoretical  Water  Consumption 975 

Leakage  of  Steam 976 

Compound  Engines. 

Advantages  of  Compounding 976 

Woolf  and  Receiver  Types  of  Engines 977 

Combined  Diagrams > 979 

Proportions  of  Cylinders  in  Compound  Engines 980 

Receiver  Space 980 

Formula  for  Calculating  Work  of  Steam 981 

Calculation  of  Diameters  of  Cylinders 982 

Triple-expansion  Engines 983 

Proportions  of  Cylinders 983 

Formulae  for  Proportioning  Cylinders 983 

Types  of  Three-stage  Expansion  Engines 985 

Sequence  of  Cranks 986 

Velocity  of  Steam  through  Passages , 986 

A  Double-tandem  Triple-expansion  Engine 986 

Quadruple-expansion  Engines 986 


CONTENTS.  XXIX 
Steam-engine  Economy. 

JrALriU 

Economic  Performance  of  Steam-engines 987 

Feed- water  Consumption  of  Different  Types 987 

Sizes  and  Calculated  Performances  of  Vertical  High-speed  Engine  988 

The  Willans  Law,  Steam  Consumption  at  Different  Loads 991 

Relative  Economy  of  Engines  under  Variable  Loads 992 

Steam  Consumption  of  Various  Sizes 992 

Steam  Consumption  in  Small  Engines 993 

Steam  Consumption  at  Various  Speeds '993 

Capacity  and  Economy  of  Steam  Fire  Engines 993 

Economy  Tests  of  High-speed  Engines 994 

Limitation  of  Engine  Speed 995 

British  High-speed  Engines 995 

Advantage  of  High  Initial  and  Low-back  Pressure 996 

Comparison  of  Compound  and  Single-cylinder  Engines 997 

Two-cylinder  and  Three-cylinder  Engines 997 

Steam  Consumption  of  Engines  with  Superheated  Steam 998 

Steam  Consumption  of  Different  Types  of  Engine 999 

The  Lentz  Compound  Engine 999 

Efficiency  of  Non-condensing  Compound  Engines 1000 

Economy  of  Engines  under  Varying  Loads 1000 

Effect  of  Water  in  Steam  on  Efficiency 1001 

Influence  of  Vacuum  and  Superheat  on  Steam  Consumption. . . .  1001 

Practical  Application  of  Superheated  Steam 1002 

Performance  of  a  Quadruple  Engine 1003 

Influence  of  the  Steam-jacket 1004 

Best  Economy  of  the  Piston  Steam  Engine 1005 

Highest  Economy  of  Pumping-engines 1006 

Sulphur-dioxide  Addendum  to  Steam-engine 1007 

Standard  Dimensions  of  Direct-connected  Generator  Sets 1007 

Dimensions  of  Parts  of  Large  Engines 1007 

Large  Rolling-mill  Engines , 1008 

Counterbalancing  Engines 1008 

Preventing  Vibrations   of  Engines * 1008 

Foundations  Embedded  in  Air 1009 

Most  Economical  Point  of  Cut-off 1009 

Type  of  Engine  used  when  Exhaust-steam  is  used  for  Heating. .  1009 

Cost  of  Steam-power 1009 

Cost  of  Coal  for  Steam-power 1010 

Power-plant  Economics 1011 

Analysis  of  Operating  Costs  of  Power-plants 1013 

Economy  of  Combination  of  Gas  Engines  and  Turbines 1014 

Storing  Steam  Heat  in  Hot  Water 1014 

Utilizing  the  Sun's  Heat  as  a  Source  of  Power 1015 

Rules  for  Conducting  Steam-engine  Tests 1015 

Dimensions  of  Parts  of  Engines. 

Cylinder. 1021 

Clearance  of  Piston 1021 

Thickness  of  Cylinder 1021 

Cylinder  Heads 1022 

Cylinder-head  Bolts 1022 

The  Piston 1023 

Piston  Packing-rings 1023 

Fit  of  Piston-rod 1024 

Diameter  of  Piston-rods 1024 

Piston-rod  Guides 1024 

The  Connecting-rod 1025 

Connecting-rod  Ends 1026 

Tapered  Connecting-rods 1026 

The  Crank-pin 1027 

Crosshead-pin  or  Wrist-pin . 1029 

The  Crank-arm 1029 

The  Shaft,  Twisting  Resistance 1030 

*    iistance  to  Bending . . , X032 


XXX  CONTENTS. 

_  PAGE 

Equivalent  Twisting  Moment 1032 

Fly-wheel  Shafts 1033 

Length  of  Shaft-bearings 1034 

Crank-shafts  with  Center-crank  and  Double-crank  Arms 1036 

Crank-shaft  with  two  Cranks  Coupled  at  90° 1037 

Crank-shaft  with  three  Cranks  at  120° 1038 

Valve-stem  or  Valve-rod 1038 

The  Eccentric 1039 

The  Eccentric-rod 1039 

Reversing-gear 1039 

Current  Practice  in  Engine  Proportions,  1897 1039 

Current  Practice  in  Steam-engine  Design,  1909 1040 

Shafts  and  Bearings  of  Engines 1042 

Calculating  the  Dimensions  Of  Bearings 1042 

Engine-frames  or  Bed-plates 1044 

Fly-wheels. 

Weight  of  Fly-wheels 1044 

Weight  of  Fly-wheels  for  Alternating-current  Units 1047 

Centrifugal  Force  in  Fly-wheels 1047 

Diameters  for  Various  Speeds 1048 

Strains  in  the  Runs 1049 

Arms  of  Fly-wheels  and  Pulleys 1050 

Thickness  of  Rims 1050 

A  Wooden  Rim  Fly-wheel 1051 

Wire- wound  Fly-wheels 1052 

The  Slide-Valve. 

Definitions,  Lap,  Lead,  etc 1052 

Sweet's  Valve-diagram , 1054 

The  Zeuner  Valve-diagram 1054 

Port  Opening,  Lead,  and  Inside  Lead 1057 

Crank  Angles  for  Connecting-rods  of  Different  Lengths 1058 

Ratio  of  Lap  and  of  Port-opening  to  Valve- travel 1058 

Relative  Motions  of  Crosshead  and  Crank 1060 

Periods  of  Admission  or  Cut-off  for  Various  Laps  and  Travels. .  1060 

Piston- valves 1061 

Setting  the  Valves  of  an  Engine 1061 

To  put  an  Engine  on  its  Center 1061 

Link-motion 1062 

The  Walschaerts  Valve-gear 1064 

Governors. 

Pendulum  or  Fly-ball  Governors 1065 

To  Change  the  Speed  of  an  Engine 1066 

Fly-wheel  or  Shaft  Governors 1066 

The  Rites  Inertia  Governor 1066 

Calculation  of  Springs  for  Shaft-governors 1066 

Condensers,  Air-pumps,  Circulating-pumps,  etc.. 

The  Jet  Condenser 1068 

Quantity  of  Cooling  Water 1068 

Ejector  Condensers 1069 

The  Barometric  Condensers 1069 

The  Surface  Condenser 1069 

Coefficient  of  Heat  Transference  in  Condensers 

The  Power  Used  for  Condensing  Apparatus 

Vacuum,  Inches  of  Mercury  and  Absolute  Pressure 

Temperatures,  Pressures  and  Volumes  of  Saturated  Air 

Condenser  Tubes 1072 

Tube-plates 1073 

Spacing  of  Tubes 1073 

Air-pump 

Area  through  Valve-seats 1°73 


CONTENTS. 

PAGE 

Work  done  by  an  Air-pump 1074 

Most  Economical  Vacuum  for  Turbines 1075 

Circulating-pump 1075 

The  Leblanc  Condenser 1076 

Feed-pumps  for  Marine  Engines 1076 

An  Evaporative  Surface  Condenser 1076 

Continuous  Use  of  Condensing  Water 1076 

Increase  of  Power  by  Condensers 1077 

Advantage  of  High  Vacuum  in  Reciprocating  Engines 1078 

The  Choice  of  a  Condenser 1078 

Cooling  Towers 1079 

Calculation  of  Air  Supply  for  Cooling  Towers 1080 

Tests  of  a  Cooling  Tower  and  Condenser 1080 

Water  Evaporated  in  a  Cooling  Tower 1080 

Weight  of  Water  Vapor  mixed  with  One  Pound  of  Air -. . . .  1081 

Evaporators  and  Distillers 1082 

Rotary  Steam  Engines — Steam  Turbines. 

Rotary  Steam  Engines 1082 

Impulse  and  Reaction  Turbines 1082 

The  DeLaval  Turbine 1082 

The  Zolley  or  Rateau  Turbine 1083 

The  Parsons  Turbine 1083 

The  Westinghouse  Double-flow  Turbine 1083 

Mechanical  Theory  of  the  Steam  Turbine 1084 

Heat  Theory  of  the  Steam  Turbine 1084 

Velocity  of  Steam  in  Nozzles 1085 

Speed  of  the  Blades 1086 

Comparison  of  Impulse  and  Reaction  Turbines 1087 

Loss  due  to  Windage 1087 

Efficiency  of  the  Machine 1087 

Steam  Consumption  of  Turbines 1088 

Effect  of  Vacuum  on  Steam  Turbines 1088 

Tests  of  Turbines 1088 

Efficiency  of  the  Rankine  Cycle 1089 

Factors  for  Reduction  to  Equivalent  Efficiency 1090 

Effect  of  Pressure,  Vacuum  and  Superheat 1090 

Steam  and  Heat  Consumption  of  the  Ideal  Engine 1091 

Westinghouse  Turbines  at  74th  St.  Station,  New  York 1092 

A  Steam  Turbine  Guarantee 1092 

Efficiency  of  a  5000-K. W.  Steam  Turbine  Generator 1092 

Comparison  of  Large  Turbines  and  Reciprocating  Engines  .....  1092 

Steam  Consumption  of  Small  Steam  Turbines 1093 

Low-pressure  Steam  Turbines 1093 

Tests  of  a  15,000-K.W.  Steam-engine  Turbine  Unit 1095 

Reduction  Gear  for  Steam  Turbines 1095 

Hot-air  Engines. 

Hot-air  or  Caloric  Engines * 1095 

Test  of  a  Hot-air  Engine , 1095 

INTERNAL,    COMBUSTION    ENGINES. 

Four-cycle  and  Two-cycle  Gas-engines 1096 

Temperatures  and  Pressures  Developed 1096 

Calculation  of  the  Power  of  Gas-engines 1097 

Pressures  and  Temperatures  at  End  of  Compression 1098 

Pressures  and  Temperature  at  Release 1099 

after  Combustion 1099 

Mean  Effective  Pressures 1099 

Sizes  of  Large  Gas-engines 1100 

Engine  Constants  for  Gas-engines 1101 

Rated  Capacity  of  Automobile  Engines 1101 

Estimate  of  the  Horse-power  of  a  Gas-engine 1101 


XXX11  CONTENTS. 

PAGE 

Oil  and  Gasoline  Engines 1101 

The  Diesel  Oil  Engine 1102 

The  De  La  Vergne  Oil  Engine 1102 

Alcohol  Engines 1102 

Ignition 1102 

Timing 1103 

Governing 1103 

Gas  and  Oil  Engine  Troubles 1103 

Conditions  of  Maximum  Efficiency 1103 

Heat  Losses  in  the  Gas-engine 1104 

Economical  Performance  of  Gas-engines 1104 

Utilization  of  Waste  Heat  from  Gas-engines 1105 

Rules  for  Conducting  Tests  of  Gas  and  Oil  Engines 1105 

LOCOMOTIVES. 

Resistance  of  Trains 1108 

Resistance  of  Electric  Railway  Cars  and  Trains 1110 

Efficiency  of  the  Mechanism  of  a  Locomotive 1111 

Adhesion 1111 

Tractive  Force 1111 

Size  of  Locomotive  Cylinders 1112 

Horse-power  of  a  Locomotive 1113 

Size  of  Locomotive  Boilers 1113 

Wootten's  Locomotive 1114 

Grate-surface,  Smokestacks,  and  Exhaust-nozzles 1115 

Fire-brick  Arches 1115 

Economy  of  High  Pressures 1116 

Leading  American  Types 1116 

Classification  of  Locomotives 1116 

Steam  Distribution  for  High  Speed 1117 

Formulae  for  Curves . 1117 

Speed  of  Railway  Trains 1118 

Performance  of  a  High-speed  Locomotive 1118 

Fuel  Efficiency  of  American  Locomotives 1119 

Locomotive  Link-motion 1119 

Dimensions  of  Some  American  Locomotives 1120 

The  Mallet  Compound  Locomotive 1120 

Indicated  Water  Consumption 1122 

Indicator  Tests  of  a  Locomotive  at  High-speed 1122 

Locomotive  Testing  Apparatus 1123 

Weights  and  Prices  of  Locomotives 1124 

Waste  of  Fuel  in  Locomotives 1 125 

Advantages  of  Compounding 1 125 

Depreciation  of  Locomotives 1125 

Average  Train  Loads 1125 

Tractive  Force  of  Locomotives,  1893  and  1905 1125 

Superheating  in  Locomotives 1126 

Counterbalancing  Locomotives 

Narrow-gauge  Railways 1127 

Petroleum-burning  Locomotives 

Fireless  Locomotives :....-....  1127 

Self-propelled  Railway  Cars 

Compressed-air  Locomotives 1128 

Air  Locomotives  with  Compound  Cylinders 1129 

SHAFTING. 

Diameters  to  Resist  Torsional  Strain 1130 

Deflection  of  Shafting 1131 

Horse-power  Transmitted  by  Shafting 1132 

Flange  Couplings 1133 

Effect  of  Cold  Rolling 1133 

Hollow  Shafts. .                                                1133 

Sizes  of  Collars  for  Shafting 1133 

Table  for  Laying  Out  Shafting , , , , , , U34 


* 

CONTENTS.  XXxiii 

^ 

PULLETS.  PAGE 

Proportions  of  Pulleys 1135 

Convexity  of  Pulleys 1136 

Cone  or  Step  Pulleys 1 136 

Method  of  Determining  Diameter^  of  Cone  Pulleys 1136 

Speeds  of  Shafts  with  Cone  Pulleys 1137 

Speeds  in  Geometrical  Progression ,  1138 

BELTING. 

Theory  of  Belts  and  Bands 1138 

Centrifugal  Tension 1139 

Belting  Practice,  Formulae  for  Belting 1139 

Horse-power  of  a  Belt  one  inch  wide 1140 

A.  F.  Nagle's  Formula 1141 

Width  of  Belt  for  Given  Horse-power , . . .  1141 

Belt  Factors 1 142 

Taylor's  Rules  for  Belting 1143 

Earth's  Studies  on  Belting 1146 

Notes  on  Belting 1146 

Lacing  of  Belts 1147 

Setting  a  Belt  on  Quarter-twist 1147 

To  Find  the  Length  of  Belt 1148 

To  Find  the  Angle  of  the  Arc  of  Contact 1148 

To  Find  the  Length  of  Belt  when  Closely  Rolled 1148 

To  Find  the  Approximate  Weight  of  Belts 1148 

Relations  of  the  Size  and  Speeds  of  Driving  and  Driven  Pulleys.  1148 

Evils  of  Tight  Belts 1149 

Sag  of  Belts 1149 

Arrangement  of  Belts  and  Pulleys 1149 

Care  of  Belts 1150 

Strength  of  Belting (. .  1150 

Adhesion,  Independent  of  Diameter rv.  1151 

Endless  Belts 1151 

Belt  Data 1151 

U.  S.  Navy  Specifications  for  Leather  Belting 1151 

Belt  Dressings 1151 

Cement  for  Cloth  or  Leather 1152 

Rubber  Belting 1152 

Steel  Belts 1152 

Chain  Drives. 

Roller  Chain  and  Sprocket  Drives 1153 

Belting  versus  Chain  Drives 1155 

Data  used  in  Design  of  Chain  Drives 1156 

Comparison  of  Rope  and  Chain  Drives 1157 

GEARING. 

Pitch,  Pitch-circle,  etc 3157 

Diametral  and  Circular  Pitch 1158 

Diameter  of  Pitch-line  of  Wheels  from  10  to  100  Teeth 1159 

Chordal  Pitch 1159 

Proportions  of  Teeth 1 159 

Gears  with  Short  Teeth 1160 

Formulae  for  Dimensions  of  Teeth 1160 

Width  of  Teeth 1161 

Proportions  of  Gear-wheels 1161 

Rules  for  Calculating  the  Speed  of  Gears  and  Pulleys 1162 

Milling  Cutters  for  Interchangeable  Gears 1162 

Forms  of  the  Teeth. 

The  Cycloidal  Tooth 1162 

The  Involute  Tooth 1165 


XXXiV  CONTENTS, 

PAGE 

Approximation  by  Circular  Arcs -. 1166 

Stub  Gear  Teeth  for  Automobiles 1167 

Stepped  Gears 1168 

Twisted  Teeth 1168 

Spiral  Gears 1168 

Worm  Gearing 1168 

The  Hindley  Worm 1169 

Teeth  of  Bevel-wheels 1169 

Annular  and  Differential  Gearing 1169 

Efficiency  of  Gearing 1170 

Efficiency  of  Worm  Gearing 1171 

Efficiency  of  Automobile  Gears 1172 

Strength  of  Gear  Teeth. 

Various  Formulae  for  Strength 1172 

Comparison  of  Formulae 1 174 

Raw-hide  Pinions 1177 

Maximum  Speed  of  Gearing 1177 

A  Heavy  Machine-cut  Spur-gear 1 178 

Frictional  Gearing 1178 

Frictional  Grooved  Gearing 1178 

Power  Transmitted  by  Friction  Drives 1178 

Friction  Clutches 1179 

Coil  Friction  Clutches 1180 

HOISTING  AND   CONVEYING. 

Working  Strength  of  Blocks 1181 

Chain-blocks 1181 

Efficiency  of  Hoisting  Tackle 1182 

Proportions  of  Hooks 1182 

Heavy  Crane  Hooks 1183 

Strength  of  Hooks  and  Shackles 1184 

Power  of  Hoisting  Engines 1184 

Effect  of  Slack  Rope  on  Strain  in  Hoisting 1186 

Limit  of  Depth  for  Hoisting 1 186 

Large  Hoisting  Records \ 1186 

Safe  Loads  for  Ropes  and  Chains 1187 

Pneumatic  Hoisting 1 187 

Counterbalancing  of  Winding-engines 1188 

Cranes. 

Classification  of  Cranes 1189 

Position  of  the  Inclined  Brace  in  a  Jib  Crane 1190 

Electric  Overhead  Traveling  Cranes 1190 

Power  Required  to  Drive  Cranes 1191 

Dimensions,  Loads  and  Speeds  of  Electric  Cranes 1191 

Notable  Crane  Installations 1192 

A  150-ton  Pillar  Crane 1192 

Compressed-air  Traveling  Cranes 1192 

Electric  versus  Hydraulic  Cranes 

Power  Required  for  Traveling  Cranes  and  Hoists 1193 

Lifting  Magnets 1193 

Telpherage 1196 

'  Coal-handling  Machinery. 

Weight  of  Overhead  Bins 1196 

Supply-pipes  from  Bins 1196 

Types  of  Coal  Elevators 1196 

Combined  Elevators  and  Conveyors 1197 

Coal  Conveyors 1 197 

Horse-power  of  Conveyors 1 198 


CONTENTS.  XXXV 

PAGE 

Bucket,  Screw,  and  Belt  Conveyors 1198 

Weight  of  Chain  and  of  Flights 1199 

Capacity  of  Belt  Conveyors 1 199 

Belt  Conveyor  Construction 1200 

Horse-power  to  Drive  Belt  Conveyors 1200 

Relative  Wearing  Power  of  Conveyor  Belts * 1200 

Pneumatic  Conveying 1201 

Pneumatic  Postal  Transmission 1201 


, 


Wire-rope  Haulage. 

Self-acting  Inclined  Plane 1202 

Simple  Engine  Plane 1203 

Tail-rope  System 1203 

Endless  Rope  System 1203 

Wire-rope  Tramways 1204 

Stress  in  Hoisting-ropes  on  Inclined  Planes 1204 

An  Aerial  Tramway  21  miles  long .. . .  1205 

Suspension  Cableways  and  Cable  Hoists 1205 

Tension  Required  to  Prevent  Wire  Slipping  on  Drums 1206 

Formulae  for  Deflection  of  a  Wire  Cable 1207 

Taper  Ropes  of  Uniform  Tensile  Strength 1208 


WIRE-ROPE  TRANSMISSION. 


Working  Tension  of  Wire  Ropes 1208 

Sheaves  for  Wire-rope  Transmission 1208 

Breaking  Strength  of  Wire  Ropes. 1209 

Bending  Stresses  of  Wire  Ropes 1209 

Horse-power  Transmitted 1210 

Diameters  of  Minimum  Sheaves 1211 

Deflection  of  the  Rope 1211 

Limits  of  Span 1212 

Long-distance  Transmission 1212 

Inclined  Transmissions 1212 

Bending  Curvature  of  Wire  Ropes 1213 

ROPE-DRIVING. 

Formulae  for  Rope-driving 1214 

Horse-power  of  Transmission  at  Various  Speeds 1215 

Sag  of  the  Rope  between  Pulleys. 1216 

Tension  on  the  Slack  Part  of  the  Rope 12*16 

Miscellaneous  Notes  on  Rope-driving .  .  , 1217 

Data  of  Manila. Transmission  Rope 1218 

Cotton  Ropes 1218 

FRICTION  AND   LUBRICATION. 

Coefficient  of  Friction 1219 

Rolling  Friction '. 1219 

Friction  of  Solids < 1219 

Friction  of  Rest , 1219 

Laws  of  Unlubricated  Friction 1219 

Friction  of  Tires  Sliding  on  Rails 1219 

Coefficient  of  Rolling  Friction 1220 

Laws  of  Fluid  Friction .' 1220 

Angles  of  Repose  of  Building  Materials 1220 

Coefficient  of  Friction  of  Journals 1220 

Friction  of  Motion 1221 

Experiments  on  Friction  of  a  Journal 1221 

Coefficients  of  Friction  of  Journal  with  Oil  Bath 1221,  1223 

Coefficients  of  Friction  of  Motion  and  of  Rest 1222 

Value  of  Anti-friction  Metals . .  1223 

Cast-iron  for  Bearings 1223 


x — ^y  CONTENTS. 

PAGE 

Friction  of  Metal  under  Steam-pressure 1223 

Morin's  Laws  of  Friction 1223 

Laws  of  Friction  of  Well-lubricated  Journals 1225 

Allowable  Pressures  on  Bearing-surfaces 1226 

Oil-pressure  in  a  Bearing 1228 

Friction  of  Car-journal  Brasses 1228 

Experiments  on  Overheating  of  Bearings 1228 

Moment  of  Friction  and  Work  of  Friction 1229 

Tests  of  Large  Shaft  Bearings 1230 

Clearance  between  Journal  and  Bearing 1230 

Allowable  Pressures  on  Bearings 1230 

Bearing  Pressures  for  Heavy  Intermittent  Loads 1231 

Bearings  for  Very  High  Rotative  Speed 1231 

Bearing  Pressures  in  Shafts  of  Parsons  Turbine 1232 

Thrust  Bearings  in  Marine  Practice 1232 

Bearings  for  Locomotives 1232 

Bearings  of  Corliss  Engines 1232 

Temperature  of  Engine  Bearings 1232 

Pivot  Bearings 1232 

The  Schiele  Curve 1232 

Friction  of  a  Flat  Pivot-bearing 1233 

Mercury-bath  Pivot 1233 

Ball  Bearings,  Roller  Bearings,  etc 1233 

Friction  Rollers 1233 

Conical  Roller  Thrust  Bearings. 1234 

The  Hyatt  Roller  Bearing 1235 

Notes  on  Ball  Bearings 1235 

Saving  of  Power  by  Use  of  Ball  Bearings 1237 

Knife-edge  Bearings 1238 

Friction  of  Steam-engines 1238 

Distribution  of  the  Friction  of  Engines 1238 

Friction  Brakes  and  Friction  Clutches. 

Friction  Brakes 1239 

Friction  Clutches 1239 

Magnetic  and  Electric  Brakes 1240 

Design  of  Band  Brakes 1240 

Friction  of  Hydaulic  Plunger  Packing 1241 

Lubrication. 

Durability  of  Lubricants 1241 

§ualifications  of  Lubricants 1242 

xamination  of  Oils 1242 

Specifications  for  Petroleum  Lubricants 1243 

Penna.  R.  R.  Specifications 1244 

Grease  Lubricants 

Testing  Oil  for  Steam  Turbines 1244 

8uantity  of  Oil  to  Run  an  Engine 

ylinder  Lubrication 1245 

Soda  Mixture  for  Machine  Tools 

Water  as  a  Lubricant 124 

Acheson's  Deflocculated  Graphite 1246 

Solid  Lubricants 1246 

Graphite,  Soapstone,  Metaline 1246 

THE  FOUNDRY. 

Cupola  Practice 1247 

Melting  Capacity  of  Different  Cupolas 1248 

Charging  a  Cupola 1248 

Improvement  of  Cupola  Practice 

Charges  in  Stove  Foundries 1250 

Foundry  Blower  Practice 1250 


CONTENTS.  XXXV11 


PAGE 

Results  of  Increased  Driving 1252 

Power  Required  for  a  Cupola  Fan 1253 

Utilization  of  Cupola  Gases 1253 

Loss  of  Iron  in  Melting 1253 

Use  of  Softeners ; . .  1253 

Weakness  of  Large  Castings 1253 

Shrinkage  of  Castings 1254 

Growth  of  Cast  Iron  by  Heating 1254 

Hard  Iron  due  to  Excessive  Silicon 1254 

Ferro  Alloys  for  Foundry  Use 1255 

Dangerous  Ferro-silicon 1255 

Quality  of  Foundry  Coke 1255 

Castings  made  in  Permanent  Cast-iron  Molds 1255 

Weight  of  Castings  from  Weight  of  Pattern 1256 

Molding  Sand 1256 

Foundry  Ladles 1257 


: 


THE  MACHINE-SHOP. 


Speed  of  Cutting  Tools 1258 

Table  of  Cutting  Speeds 1258 

Spindle  Speeds  of  Lathes 1259 

Rule  for  Gearing  Lathes 1259 

Change-gears  for  Lathes 1260 

Quick  Change  Gears 1260 

Metric  Screw-threads 1261 

Cold  Chisels 1261 

Setting  the  Taper  in  a  Lathe 1261 

Lubricants  for  Lathe  Centers 1261 

Taylor's  Experiments  on  Tool  Steel 1261 

Proper  Shape  of  Lathe  Tool 1261 

Forging  and  Grinding  Tools 1263 

Best  Grinding  Wheel  for  Tools 1263  -~- 

Chatter « 1264 

Use  of  Water  on  Tool 1264 

Interval  between  Grindings 1264 

Effect  of  Feed  and  Depth  of  Cut  on  Speed 1264 

Best  High  Speed  Tool  Steel — Heat  Treatment 1265 

Table,  Cutting  Speeds  of  Taylor- White  Tools 1266 

Best  Method  of  Treating  Tools  in  Small  Shops 1268 

Quality  of  Different  Tool  Steels 1268 

Parting  and  Thread  Tools 1268 

Durability  of  Cutting  Tools 1268 

Economical  Cutting  Speeds 1268  -^ 

New  High  Speed  Steels,  1909 1269 

Stellite 1269 

Planer  Work 1270-1275 

Cutting  and  Return  Speeds  of  Planers 1270 

Power  Required  for  Planing 1270 

Time  Required  for  Planing 1271 

Standard  Planer  Tools 1271-1275 

Milling  Machine  Practice <, 1275-1284 

Forms  of  Milling  Cutters 1275 

Number  of  Teeth  in  Milling  Cutters 1276 

Keyways  in  Milling  Cutters 1277 

Power  Required  for  Milling 1278 

Modern  Milling  Practice,  1914 1279 

Milling  wiJi  or  against  the  Feed 1280 

Lubricant  for  Milling  Cutters •:.  . .  1281 


Typica 
High-s 


Jigh-speed  Milling 1282 

Limiting  Factors  of  Milling  Practice 1283 

Speeds  and  Feeds  for  Gear  Cutting 1284 

Drills  and  Drilling 1285-1290 

Forms  of  Drills 1285 

Drilling  Compounds 1286 


XXXV111  CONTENTS. 

1>AGE 

Twist  Drill  and  Steel  Wire  Gages ...;>...,.. 1286 

Power  Required  to  Drive  Drills 1286,  1287 

Feeds  and  Speeds  of  Drills 1288 

Extreme  Results  with  Drills ; 1289 

Experiments  on  Twist  Drills 1289 

Cutting  Speeds  for  Tapping  and  Threading 1290 

Sawing  Metals 1291 

Case-hardening,  Cementation,  Harvey izing 1291 

Change  of  Shape  due  to  Hardening  and  Tempering 1291 

Power  Required  for  Machine  Tools. 

Resistance  Overcome  in  Cutting  Metal :. .  1292 

Power  Required  to  Run  Lathes 1292-1295 

Sizes  of  Motors  for  Machine  Tools 1294-1298 

Horse-power  Constants  for  Cutting  Metals 1299 

Pulley  Diameters  for  Motors 1300 

Geared  Connections  for  Motors,  Table 1301 

Motor  Requirements  for  Planers 1302 

Tests  on  a  Motor-driven  Planer 1303 

Power  Required  for  Wood-working  Machinery 1303 

Power  Required  to  Drive  Shafting 1305 

Power  Required  to  Drive  Machines  in  Groups 1305 

Machine  Tool  Drives,  Speeds  and  Feeds 1307 

Geometrical  Progression  of  Speeds  and  Feeds 1307 

Methods  of  Driving  Machine  Tools 1307 

Abrasive  Processes. 

The  Cold  Saw 1309 

Reese's  Fusing-disk 1309 

Cutting  Stone  with  Wire 1309 

The  Sand-blast 1309 

Polishing  and  Buffing '. 1310 

Laps  and  Lapping 1310 

Emery-wheels 131 1-1317 

Artificial  Abrasives 1313 

Mounting  Grinding  Wheels,  Safety  Devices 1314 

Grinding  as  a  Substitute  for  Finish  Turning 1317 

Grindstones 1317 

Various  Tools  and  Processes. 

Taper  Bolts,  Pins,  Reamers,  etc 1318 

Morse  Tapers 1319 

Jarno  Taper 1319 

Tap  Drills 1320 

Taper  Pins 1321 

T-slots,  T-bolts  and  T-nuts 1321 

Punches  and  Dies,  Presses,  etc 1321 

Punch  and  Die  Clearances > . .  1321 

Kennedy's  Spiral  Punch 1322 

.  Sizes  of  Blanks  Used  in  the  Drawing  Press 1322 

Pressure  Obtained  by  the  Drop  Press 1322 

Flow  of  Metals 1323 

Fly-wheels  for  Presses,  Punches,  Shears,  etc. 1323 

Forcing,  Shrinking,  and  Running  Fits 1324 

Pressures  for  Mounting  Wheels  and  Crank  Pins 1324 

Fits  for  Machine  Parts 1325 

Running  Fits 1325 

Shop  Allowances  for  Electrical  Machinery 

Pressure  Required  for  Press  Fits 

Stresses  due  to  Force  and  Shrink  Fits 1326 

Force  Required  to  Start  Force  and  Shrink  Fits 1327 

Formulae  for  Flat  and  Square  Keys 1328 


CONTENTS.  XXXIX 

PAGE 

Keys  of  Various  Forms 1328-1331 

Depth  of  Key  Seats 1329 

Gib  Keys 1332 

Holding  Power  of  Keys  and  Set  Screws 1332 

DYNAMOMETERS. 

Traction  Dynamometers 1333 

The  Prony  Brake 1333 

The  Alden  Dynamometer 1334 

Capacity  of  Friction-brakes 1334 

Transmission  Dynamometers 1335 

ICE  MAKING  OR  REFRIGERATING-MACHINES. 

Operations  of  a  Refrigerating-Machine 1336 

Pressures,  etc.,  of  Available  Liquids 1337 

Properties  of  Sulphur  Dioxide  Gas 1338 

Properties  of  Ammonia 1339,  1340 

Solubility  of  Ammonia 1341 

Properties  of  Saturated  Vapors 1341 

Heat  Generated  by  Absorption  of  Ammonia 1341 

Cooling  Effect,  Compressor  Volume  and  Power  Required,  with 

Different  Cooling  Agents 1341 

Ratios  of  Condenser,  Mean  Effective,  and  Vaporizer  Pressures .  .  1342 

Properties  of  Brine  used  to  absorb  Refrigerating  Effect 1343 

Chloride-of-calcium  Solution 1343 

Ice-melting  Effect 1344 

Ether-machines 1344 

Air-machines 1344 

Carbon  Dioxide  Machines 1344 

Methyl  Chloride  Machines 1345 

Sulphur-dioxide  Machines 1345 

Machines  Using  Vapor  of  Water 1345 

Ammonia  Compression-machines 1345 

Dry,  Wet  and  Flooded  Systems 1345 

Ammonia  Absorption-machines 1346 

Relative  Performance  of  Compression  and  Absorption  Machines  1346 

Efficiency  of  a  Refrigerating-machine 1347 

Diagrams  of  Ammonia  Machine  Operation 1348 

Cylinder-heating 1349 

Volumetric  Efficiency 1349 

Pounds  of  Ammonia  per  Ton  of  Refrigeration 1350,  1351 

Mean  Effective  Pressure,  and  Horse-power 1350 

The  Voorhees  Multiple  Effect  Compressor 1350 

Size  and  Capacities  of  Ammonia  Machines ,  .  .  .  .  1352 

Piston  Speeds  and  Revolutions  per  Minute 1353 

Condensers  for  Refrigera  ting-machines 1353 

Cooling  Tower  Practice  in  Refrigerating  Plants 1354 

Test  Trials  of  Refrigerating-machines 1355 

Comparison  of  Actual  and  Theoretical  Capacity 1355 

Performance  of  Ammonia  Compression-machines 1 356 

Economy  of  Ammonia  Compression-machines 1357 

Form  of  Report  of  Test 1358 

Temperature  Range 1359 

Metering  the  Ammonia 1359 

Performance  of  Ice-making  Machines 1359 

Performance  of  a  75-ton  Refrigerating-machine 1361-1363 

Ammonia  Compression-machine,  Results  of  Tests '.  .  1364 

Performance  of  a  Single-acting  Ammonia  Compressor 1364 

Performance  of  Ammonia  Absorption-machine 1364 

Means  for  Applying  the  Cold 1365 

Artificial  Ice-manufacture , Ib66 

Test  of  the  New  York  Hygeia  Ice-making  Plant 1367 

An  Absorption  Evaporator  Ice-making  System 1367 

Ice-making  with  Exhaust  Steam 1367 


Xl  CONTENTS. 

PAGE 

Tons  of  Ice  per  Ton  of  Coal 1367 

Standard  Ice  Cans  or  Molds 1368 

Cubic  Feet  of  Insulated  Space  per  Ton  Refrigeration 1368 

MARINE  ENGINEERING. 

Rules  for  Measuring  and  Obtaining  Tonnage  of  Vessels 1368 

The  Displacement  of  a  Vessel 1369 

Coefficient  of  Fineness 1369 

Coefficient  of  Water-line 1369 

Resistance  of  Ships 1369 

Coefficient  of  Performance  of  Vessels 1370 

Defects  of  the  Common  Formula  for  Resistance 1370 

Rankine's  Formula 1370 

Empirical  Equations  for  Wetted  Surface 1371 

E.  R.  Mumford's  Method 1371 

Dr.  Kirk's  Method 1372 

To  find  the  I.H.P.  from  the  Wetted  Surface. 1372 

Relative  Horse-power  required  for  Different  Speeds  of  Vessels .  .  1373 

Resistance  per  Horse-power  for  Different  Speeds 1373 

Estimated  Displacement,  Horse-power,  etc.,  of  S team- vessels. .  .  1374 

Speed  of  Boats  with  Internal  Combustion  Engines 1374 

Data  of  Ships  of  Various  Types 1376 

Relation  of  Horse-power  to  Speed 1376 

The  Screw-propeller. 

Pitch  and  Size  of  Screw 1377 

Propeller  Coefficients 1378 

Efficiency  of  the  Propeller 1379 

Pitch-ratio  and  Slip  for  Screws  of  Standard  Form 1379 

Table  for  Calculating  Dimensions  of  Screws 1380 

Marine  Practice. 

Comparison  of  Marine  Engines,  1872,  1881,  1891,  1901 1380 

Turbines  and  Boilers  of  the  "  Lusitania" 1381 

Performance  of  the  "Lusitania,"  1908 1381 

Dimensions  and  Performance  of  Notable  Atlantic  Steamers.  . .  . 

Relative  Economy  of  Turbines  and  Reciprocating  Engines 1382 

Reciprocating  Engines  with  a  Low-pressure  Turbine 1383 

The  Paddle-wheel. 

Paddle-wheels  with  Radial  Floats 1383 

Feathering  Paddle-wheels 1383 

Efficiency  of  Paddle-wheels 1384 

Jet  Propulsion. 

Reaction  of  a  Jet 1384 

CONSTRUCTION  OF  BUILDINGS. 

Foundations. 

Bearing  Power  of  Soils 1385 

Bearing  Power  of  Piles 1386 

Safe  Strength  of  Brick  Piers 1386 

Thickness  of  Foundation  Walls 1386 

Masonry. 

Allowable  Pressures  on  Masonry 1386 

Crushing  Strength  of  Concrete 1386 

Reinforced  Concrete 1386 


CONTENTS.  Xll 

Beams  and  Girders.  PAGE 

Safe  Loads  on  Beams 1387 

Safe  Loads  on  Wooden  Beams 1387 

Maximum  Permissible  Stresses  in  Structural  Materials 1388 

Walls. 

Thickness  of  Walls  of  Buildings 1388 

Walls  of  Warehouses,  Stores,  Factories,  and  Stables 1388 

Floors,  Columns  and  Posts. 

Strength  of  Floors,  Roofs,  and  Supports 1389 

Columns  and  Posts 1389 

Fireproof  Buildings 1389 

Iron  and  Steel  Columns 1389 

Lintels,  Bearings,  and  Supports 1390 

Strains  on  Girders  and  Rivets 1390 

Maximum  Load  on  Floors 1390 

Strength  of  Floors 1391 

Maximum  Spans  for  1,  2  and  3  inch  Plank 1392 

Mill  C9lumns .  , 1393 

Safe  Distributed  Loads  on  Southern-pine  Beams 1393 

Approximate  Cost  of  Mill  Buildings 1394 

ELECTRICAL  ENGINEERING. 

C.  G.  S.  System  of  Physical  Measurement 1396 

Practical  Units  used  in  Electrical  Calculations 1396 

Relations  of  Various  Units 1397 

Units  of  the  Magnetic  Circuit 1398 

Equivalent  Electrical  and  Mechanical  Units 1399 

Permeability 1400 

logics  between  Flow  of  Water  and  Electricity 1400 

Electrical  Resistance. 

Laws  of  Electrical  Resistance 1400 

Electrical  Conductivity  of  Different  Metals  and  Alloys 1401 

Conductors  and  Insulators 1402 

Resistance  Varies  with  Temperature : 1402 

Annealing 1402 

Standard  of  Resistance  of  Copper  Wire 1402 

Wire  Table,  Standard  Annealed  Copper 1404 

Direct  Electric  Currents. 

Ohm's  Law 1406 

Series  and  Parallel  or  Multiple  Circuits 1406 

Resistance  of  Conductors  in  Series  and  Parallel 1407 

Internal  Resistance 1408 

Power  of  the  Circuit 1408 

Electrical,  Indicated,  and  Brake  Horse-power 1408 

Heat  Generated  by  a  Current 1408 

Heating  of  Conductors 1409 

Heating  of  Coils 1409 

Fusion  of  Wires 1409 

Allowable  Carrying  Capacity  of  Copper  Wires : .  1410 

Underwriters'  Insulation 1410 

Electric  Transmission,  Direct-Currents. 

Drop  of  Voltage  in  Wires  Carrying  Allowed  Currents 1410 

Section  of  Wire  Required  for  a  Given  Current 1410 

Weight  of  Copper  for  a  Given  Power 1411 


Xlii  CONTENTS. 

PAGE 

Short-circuiting 1411 

Economy  of  Electric  Transmission 1411 

Efficiency  of  Electric  Systems 1412 

Wire  Table  for  110,  220,  500,  1000,  and  2000  volt  Circuits 1413 

Resistances  of  Pure  Aluminum  Wire 1414 

Electric  Railways. 

Schedule  Speeds,  Miles  per  Hour 1414 

Train  Resistance 1415 

Rates  of  Acceleration 1415 

Safe  Maximum  Speed  on  Curves 1416 

Electric  Resistance  of  Rails  and  Bonds 1416 

Electric  Locomotives 1416 

Efficiencies  of  Distributing  Systems 1417 

Steam  Railroad  Electrifications 1418 

Electric  Welding. 

Arc  Welding 1419 

Data  of  Electric  Welding  in  Railway  Shops 1419 

Resistance  Welding 1419 

Cost  of  Welding 1420 

Electric  Heaters. 

Elementary  Form  of  Heater 1420 

Relative  Efficiency  of  Electric  and  Steam  Heating 1421 

Heat  Required  to  Warm  and  Ventilate  a  Room 1421 

Domestic  Heating 1421 

Electric  Furnaces. 

Arc  Furnaces  and  Resistance  Furnaces 1422 

Uses  of  Electric  Furnaces 1423 

Electric  Smelting  of  Pig-iron 1424 

Ferro-alloys 1424 

Non-ferrous  Metals 1424 

Electric  Batteries. 

Primary  Batteries 1425 

Description  of  Storage-batteries  or  Accumulators 1425 

Rules  for  Care  of  Storage-batteries 1426 

Efficiency  of  a  Storage  Cell 1427 

Uses  of  Storage-batteries '. 1427 

Edison  Alkaline  Battery 1428 

Electrolysis 1428 

Electro-chemical  Equivalents 1429 

The  Magnetic  Circuit. 

Lines  and  Loops  of  Force 1430 

Values  of  B  and  H 1431 

Tractive  or  Lifting  Force  of  a  Magnet 1431 

Determining  the  Polarity  of  Electro-magnets 

Determining  the  Direction  of  a  Current 1432 

Dynamo-electric  Machines. 

Rating  of  Generators  and  Motors 1432 

Temperature  Limitations  of  Capacity 143£ 

Methods  of  Determining  Temperatures 143 

Temperature  Limits  of  Hottest  Spot 

Moving  Force  of  a  Dynamo-electric  Machine 1435 


CONTENTS.  xliii 

PAGE 

Torque  of  an  Armature 1435 

Torque,  Horse-power  and  Revolutions 1436 

Electro-motive  Force  of  the  Armature  Circuit 1436 

Strength  of  the  Magnetic  Field 1436 

Direct-Current  Generators. 

Series-,  Shunt-  and  Compound- wound 1437 

Commutating  Pole  Machines 1438 

Parallel  Operation 1439 

Three- Wire  System 1439 

Alternating  Currents. 

Maximum,  Average  and  Effective  Values 1440 

Frequency 1440 

Inductance 1440 

Capacity 1440 

Power  Factor 1440 

Reactance,  Impedance,  Admittance 1441 

Skin  Effect 1442 

Ohm's  Law  Applied  to  Alternating  Current  Circuits 1442 

Impedance  Polygons 1442 

Self-inductance  of  Lines  and  Circuits 1446 

Capacity  of  Conductors 1446 

Single-phase  and  Polyphase  Currents t  1446 

Measurement  of  Power  in  Polyphase  Circuits 1447 

Alternating  Current  Generators. 

Synchronous  Generators 1448 

Rating 1448 

Efficiency 1448 

Regulation ; 1449 

Rating  of  a  Generator  Unit 1449 

Windings 1449 

Voltages 1450 

Parallel  Operation 1450 

Exciters 1450 

Transformers. 

Primary  and  Secondary 1451 

Voltage  Ratio 1451 

Rating 1451 

Efficiency 1451 

Connections 1452 

Auto  Transformers 1453 

Constant-Current  Transformers . , 1453 

Synchronous  Converters. 

Description 1453 

Effective  E.M.F.  between  Collector  Rings 1454 

Voltage  Regulation 1455 

Starting  Synchronous  Converters 1455 

Motor-Generators. 

Balancers 1456 

Boosters 1456 

Dynamotors 1457 

Frequency  Changers 1457 

Mercury  Arc  Rectifier 1457 


xliv  CONTENTS. 

Alternating'Current  Circuits. 

PAGfi 

Calculation  of  Alternating  Current  Circuits 1457 

Relative  Weight  of  Copper  Required  in  Different  Systems .....  1459 

Rule  for  Size  of  Wires  for  Three-phase  Transmission  Lines 1459 

Notes  on  High-tension  Transmission 1459 

Voltages  Advisable  for  Various  Line  Lengths 1460 

Line  Spacing 1460 

Size  of  Line  Conductors 1460 

A  135,000- volt  Three-phase  Transmission  System 1461 

Electric  Motors. 

Classification  of  Motors 1461 

Characteristics  of  Motors 1461 

Series  Motor 1461 

Speed  Control  of  Motors 1462 

Shunt  IMotor 1462 

Compound  Motor 1462 

Induction  Motor;  Squirrel-cage  Motor 1463 

Multi-speed  Induction  Motors 1463 

Synchronous  Motors 1463 

Single-phase  Series  Motor 1464 

Repulsion  Induction  Motor 1464 

Reversible  Repulsion  Motor 1464 

Variable-speed  Repulsion  Motor 1464 

Motor  Applications. 

Pumps 1464 

Fans 1465 

Air  Compressors 1465 

Hoists .  . 1465 

Machine  Tools 1466 

Motors  for  Machine  Tools 1467 

Illumination — Electric  and  Gas  Lighting. 

Illumination 1468 

Terms,  Units,  Definitions .... 1468 

Relative  Color  Values  of  Illuminants 1469 

Relation  of  Illumination  to  Vision •. . .  1469 

Types  of  Electric  Lamps 1470 

Street  Lighting 1470 

Illumination  by  Arc  Lamps  at  Different  Distances 1471 

Data  of  Some  Arc  Lamps 1471 

Relative  Efficiency  of  Illuminants 1472 

Characteristics  of  Tungsten  Lamps 1473 

Interior  Illumination 1473 

Quantity  of  Electricity  or  Gas  Required  for  Illuminating 1474 

Standard  Units;  Mazda  and  Welsbach 1475 

Cost  of  Electric  Lighting 1475 

Recent  Street  Lighting  Installations 1476 

Symbols  Used  in  Electric  Diagrams 1477 


NAMES  AND  ABBREVIATIONS  OF  PERIODICALS  AND 
TEXT -BOOKS  FREQUENTLY  REFERRED  TO  IN 
THIS  WORK. 


Am.  Mach.     American  Machinist. 

App.  Cyl.  Mech.     Appleton's  Cyclopaedia  of  Mechanics,  Vols.  I  and  II. 

Bull.  I.  &  S.  A.     Bulletin  of  the  American  Iron  and  Steel  Association. 

Burr's  Elasticity  and  Resistance  of  Materials. 

Clark,  R.  T.  D.  D.  K.  Clark's  Rules,  Tables,  and  Data  for  Mechanical 
Engineers. 

Clark,  S.  E.     D.  K.  Clark's  Treatise  on  the  Steam-Engine. 

Col.  Coll.  Qly.     Columbia  College  Quarterly. 

El.  Rev.     Electrical  Review. 

El.  World.     Electrical  World  and  Engineer. 

Engg.     Engineering  (London). 

Eng.  News.     Engineering  News. 

Eng.  Rec.     Engineering  Record. 

Engr.     The  Engineer  (London). 

Fairbairn's  Useful  Information  for  Engineers. 

Flynn's  Irrigation  Canals  and  Flow  of  Water. 

Indust.  Eng.     Industrial  Engineering. 

Jour.  A.  C.  I.  W.  Journal  of  American  Charcoal  Iron  Workers' 
Association. 

Jour.  Ass.  Eng.  Soc.  Journal  of  the  Association  of  Engineering 
Societies. 

Jour.  F.  I.     Journal  of  the  Franklin  Institute. 

Lanza's  Applied  Mechanics. 

Machy.     Machinery. 

Merriman's  Strength  of  Materials. 

Modern  Mechanism.  Supplementary  volume  of  Appleton's  Cyclo- 
paedia of  Mechanics. 

Peabody's  Thermodynamics. 

Proc.  A.  S.  H.  V.  E.  Proceedings.  Am.  Soc'y  of  Heating  and  Ventilat- 
ing Engineers. 

Proc.  A.  S.  T.  M.     Proceedings  Amer.  Soc'y  for  Testing  Materials. 

Proc.  Inst.  C.  E.     Proceedings  Institution  of  Civil  Engineers  (London). 

Proc.  Inst.  M.  E.  Proceedings  Institution  of  Mechanical  Engineers 
(London) . 

Proceedings  Engineers'  Club  of  Philadelphia. 

Rankine,  S.  E.     Rankine's  The  Steam  Engine  and  other  Prime  Movers. 

Rankine's  Machinery  and  Millwork. 

Rankine,  R.  T.  D.     Rankine's  Rules,  Tables,  and  Data. 

Reports  of  U.  S.  Iron  and  Steel  Test  Board. 

Reports  of  U.  S.  Testing  Machine  at  Watertown,  Massachusetts. 

Rontgen's  Thermodynamics. 

Seaton's  Manual  of  Marine  Engineering. 

Hamilton  Smith,  Jr.'s  Hydraulics. 

Stevens  Indicator. 

Thompson's  Dynamo-electric  Machinery. 

Thurston's  Manual  of  the  Steam  Engine.  « 

Thurston's  Materials  of  Engineering. 

Trans.  A.  I.  E.  E.  Transactions  American  Institute  of  Electrical 
Engineers. 

Trans.  A.  I.  M.  E.  Transactions  American  Institute  of  Mining 
Engineers. 

Trans.  A.  S.  C.  E.     Transactions  American  Society  of  Civil  Engineers. 

Trans.  A.  S.  M.  E.  Transactions  American  Society  of  Mechanical 
Engineers. 

Trautwine's  Civil  Engineer's  Pocket  Book. 

The  Locomotive  (Hartford,  Connecticut). 

Unwin's  Elements  of  Machine  Design. 

Weisbach's  Mechanics  of  Engineering, 

Wood's  Resistance  of  Materials. 

Wood's  Thermodynamics,   _, 


MATHEMATICS. 


Greek  Letter. 


a  Alpha 

j8  Beta 

y  Gamma 

8  Delta 

e  Epsilon 

C  Zeta 


Eta 

N     v 

Nu 

T 

T    Tan 

9  Theta 

H     £ 

Xi 

Y 

v   Upsilon 

Iota 

0      o 

Omicron 

$ 

<t>   Phi 

Kappa 
Lambda 

II       7T 
P        P 

Pi 

Kho 

X 

Y   Chi 
$   Psi 

Mu 

2     as 

Sigma 

O 

w   Omega 

Arithmetical  and  Algebraical  Signs  and  Abbreviations* 


+  plus  (addition). 
+  positive. 
—  minus  (subtraction). 
-  negative. 
±  plus  or  minus. 
T  minus  or  plus. 
=  equals. 
X  multiplied  by. 
ab  or  a.b  =  a  X  b. 
-5-  divided  by. 
/   divided  by. 

2  _«/6  _««.».     15-16  =  if  - 
0.2 -£;  0.002 -jJL. 

V  square  root. 
^  cube  root. 
M  4th  root. 

:  is  to,  ::  so  is,  :  to  (proportion). 
2  :  4  ::  3  :  6,  2  is  to  4  as  3  is  to  6. 
:  ratio;  divided  by. 
2  :  4,  ratio  of  2  to  4  =  2/4. 
.*.  therefore. 
>  greater  than. 
<  less  than. 
D  square. 
O  round. 

0  degrees,  arc  or  thermometer. 
'  minutes  or  feet. 
"  seconds  or  inches. 

"'  accents  to  distinguish  letters, 

as  a',  a",  a'". 
<*!•  «2,  03,  ab,  etc,  read  a  sub  1,  a  sub 

ft,  etc. 


on) 


-  parenthesis,  braclr^ts, 


braces,     vinculum ;     denoting 
that  the  numbers  enclosed  are 
to  be  taken  together;  as, 
(a  +  b)c  =  4  +  3  X  5  =  35, 

a2,  a3,  a  squared,  a  cubed. 

an,  a  raised  to  the  nth  power. 


109  =  10    to     the    9th     power  = 

1,000,000,000. 
sin  a  =  the  sine  of  a. 
sin"1  a  =  the  arc  whose  sine  is  a. 

sin  a-»  =    — ^ — 

sin  a 

log  =  logarithm. 

loge  or  hyp  log  =  hyperbolic  loga- 
rithm. 
%  per  cent. 
A  angle. 


,L  right  angle. 

JL  perpendicular  to. 

sin,  sine. 

cos,  cosine. 

tan,  tangent. 

sec,  secant. 

versin,  versed  sine. 

cot,  cotangent. 

cosec,  cosecant. 

covers,  co-versed  sine. 

In  Algebra,  the  first  letters  of 
the  alphabet,  a,  b,  c,  d,  etc.,  are 
generally  used  to  denote  known 
quantities,  and  the  last  letters, 
w,  x,  y,  z,  etc.,  unknown  quantities. 

Abbreviations    and    Symbols    com- 
monly used, 
d,  differential  (in  calculus). 

,  integral  (in  calculus). 


,  integral  between  limits  a  and  b. 


A,  delta,  difference. 

2,  sigma,  sign  of  summation. 

n,  pi,    ratio    of   circumference   of 

circle  to  diameter  =  3.14159. 
g,  acceleration   due  to  gravity  = 

32.16  ft.  per  second  per  second. 

Abbreviations    frequently    used    in 

this  Book. 

L.,  1.,  length  in  feet  and  inches. 
B.,  b.,  breadth  in  feet  and  inches. 
D.,  d.,  depth  or  diameter. 
H.,  h.,  height,  feet  and  inches. 
T.,  t.,  thickness  or  temperature. 
V.,  v.,  velocity. 
F.,  force,  or  factor  of  safety, 
f.,  coefficient  of  fricti9n. 
E.,  coefficient  of  elasticity. 
11.,  r.,  radius. 
W.,  w.,  weight. 
P.,  p.,  pressure  or  load. 
H.P.,  horse-power. 
I.H.P.,  indicated  horse-power. 
B.H.P.,  brake  horse-power, 
h.  p.,  high  pressure, 
i.  p.,  intermediate  pressure. 
I.  p.,  low  pressure. 
A.W.G.,     American    Wire    Gauge 

(Brown  &  Sharpe). 
B.W.G.,  Birmingham  Wire  Gauge. 
r.  p.  m.,  or  revs,  per  min..  revolu- 
tions per  minute. 
Q.  =*  quantity,  or  volume. 


ARITHMETIC. 


.    ARITHMETIC. 

The  user  of  this  book  is  supposed  to  have  had  a  training  in  arithmetic  as 
well  as  in  elementary  algebra.  Only  those  rules  are  given  here  which  are 
apt  to  be  easily  forgotten. 

GREATEST  COMMON  MEASURE,  OR  GREATEST 
COMMON  DIVISOR  OF  TWO  NUMBERS. 

Rule.  —  Divide  the  greater  number  by  the  less;  then  divide  the  divisor 
by  the  remainder,  and  so  on,  dividing  always  the  last  divisor  by  the  last 
remainder,  until  there  is  no  remainder,  and  the  last  divisor  is  the  greatest 
common  measure  required. 


LEAST  COMMON  MULTIPLE  OF  TWO  OR  MORE 
NUMBERS. 

Rule.  —  Divide  the  given  numbers  by  any  number  that  will  divide  the 
greatest  number  of  them  without  a  remainder,  and  set  the  quotients  with 
the  undivided  numbers  in  a  line  beneath. 

Divide  the  second  line  as  before,  and  so  on,  until  there  are  no  two  num- 
bers that  can  be  divided;  then  the  continued  product  of  the  divisors,  last 
quotients,  and  undivided  numbers  will  give  the  multiple  required. 

FRACTIONS. 

To  reduce  a  common  fraction  to  its  lowest  terms.  —  Divide  both 
terms  by  their  greatest  common  divisor:  39/52  =  3/4. 

To  change  an  improper  fraction  to  a  mixed  number.  —  Divide  the 
numerator  by  the  denominator;  the  quotient  is  the  whole  number,  and 
the  remainder  placed  over  the  denominator  is  the  fraction:  39/4  =  93/4. 

To  change  a  mixed  number  to  an  improper  fraction.  —  Multiply 
the  whole  number  by  the  denominator  of  the  fraction;  to  the  product  add 
the  numerator;  place  the  sum  over  the  denominator:  17/g  =  i5/8. 

To  express  a  whole  number  in  the  form  of  a  fraction  with  a  given 
denominator.  —  Multiply  the  whole  number  by  the  given  denominator, 
and  place  the  product  over  that  denominator:  13  =  39/3. 

To  reduce  a  compound  to  a  simple  fraction,  also  to  multiply 
fractions.  —  Multiply  the  numerators  together  for  a  new  numerator  and 
the  denominators  together  for  a  new  denominator: 

2.4       8      .       2^4       8 
3°f  3  =  9'  alS°    3X3  =  9' 

To  reduce  a  complex  to  a  simple  fraction.  —  The  numerator  and 
denominator  must  each  first  be  given  the  form  of  a  simple  fraction;  then 
multiply  the  numerator  of  the  upper  fraction  by  the  denominator  of  the 
lower  for  the  new  numerator,  and  the  denominator  of  the  upper  by  the 
numerator  of  the  lower  for  the  new  denominator: 

7/8    =  7/8  =  28  =  1 
l3/4        7/4        56        2* 

To  divide  fractions.  —  Reduce  both  to  the  form  of  simple  fractions, 
Invert  the  divisor,  and  proceed  as  in  multiplication: 

3  35       34       12       3 

4  +1V4  -5  +  4~  4X5~20-  5' 

Cancellation  of  fractions.  —  In  compound  or  multiplied  fractions, 
divide  any  numerator  and  any  denominator  by  any  number  which  will 
divide  them  both  without  remainder,  striking  out  the  numbers  thus 
divided  and  setting  down  the  quotients  in  their  stead. 

To  reduce  fractions  to  a  common  denominator.  —  Reduce  each 
fraction  to  the  form  of  a  simple  fraction;  then  multiply  each  numerator 


DECIMALS. 


fcy  all  the  denominators  except  its  own  for  the  new  numerator,  and  all 
the  denominators  together  for  the  common  denominator: 


— , 

42* 


14  f 
42* 


IS 
42* 


To  add  fractions.  —  Reduce  them  to  a  common  denominator,  then 
add  the  numerators  and  place  their  sum  over  the  common  denominator: 


21  +  14  4-  18 
42 


53 
43 


To  subtract  fractions.  —  Reduce  them  to  a  common  denominator, 
subtract  the  numerators  and  place  the  difference  over  the  common  denom- 
inator: 

1  _  3       7-6       J_ 

2  7  ~      14      "  14 


DECIMALS. 

To  add  decimals.  —  Set  down  the  figures  so  that  the  decimal  points 
are  one  above  the  other,  then  proceed  as  in  simple  addition:  18.75'  4-  0.012 
=  18.762. 

To  subtract  decimals.  —  Set  down  the  figures  so  that  the  decimal 
points  are  one  above  the  other,  then  proceed  as  in  simple  subtraction: 
18.75  -  0.012  =  18.738. 

To  multiply  decimals.  —  Multiply  as  in  multiplication  of  whole  num- 
bers, then  point  off  as  many  decimal  places  as  there  are  in  multiplier  and 
multiplicand  taken  together:  1.5  X  0.02  =  .030  =  0.03. 

To  divide  decimals. —  Divide  as  in  whole  numbers,  and  point  off  in 
the  quotient  as  many  decimal  places  as  those  in  the  dividend  exceed  those 
in  the  divisor.  Ciphers  must  be  added  to  the  dividend  to  make  its  decimal 
places  at  least  equal  those  in  the  divisor,  and  as  many  more  as  it  is  desired 
to  have  in  the  quotient:  1.5  -J-  0.25  =  6.  0.1  -i-  0.3  =  0.10000  -i-  0.3 
=  0.3333  +. 

Decimal  Equivalents  of  Fractions  of  One  Inch. 


1-64 

.015625 

17-64 

.265625 

33-64 

.515625 

49-64 

.765625 

1-32 

.03125 

9-32 

.28125 

17-32 

.53125 

25-32 

.78125 

3-64 

.046875 

19-64 

.296875 

35-64 

.546875 

51-64 

.796875 

1-16 

.0625 

5-16 

.3125 

9-16 

.5625 

13-16 

.8125 

5-64 

.078125 

21-64 

.328125 

37-64 

.578125 

53-64 

.828125 

3-32 

.09375 

11-32 

.34375 

19-32 

.59375 

27-32 

.84375 

7-64 

.109375 

23-64 

.359375 

39-64 

.609375 

55-64 

.859375 

1-8 

.125 

3-8 

.375 

5-8 

.625 

7-8 

.875 

9-64 

.140625 

25-64 

.390625 

41-64 

.640625 

57-64 

.890625 

5-32 

.15625 

13-32 

.40625 

21-32 

.65625 

29-32 

.90625 

11-64 

.171875 

27-64 

.421875 

43-64 

.671875 

59-64 

.921875 

3-16 

.1875 

7-16 

.4375 

11-16 

.6875 

15-16 

.9375 

13-64 

.203125 

29-64 

.453125 

45-64 

.703125 

61-64 

.953123 

7-32 

.21875 

15-32 

.46875 

23-32 

,71875 

31-32 

.96875 

15-64 

.234375 

31-64 

.484375 

47-64 

.734375 

63-64 

.984375 

1-4 

.25 

1-3 

.50 

3-4 

.75 

1 

1. 

To  convert  a  common  fraction  into  a  decimal.  —  Divide  the  nume- 
rator by  the  denominator,  adding  to  the  numerator  as  many  ciphers 
prefixed  by  a  decimal  point  as  are  necessary  to  give  the  number  of  decimal 
places  desired  in  the  result:  1/3  =  1.0000  •*•  3  =  0.3333  +. 

To  convert  a  decimal  into  a  common  fraction.  —  Set  down  the 
decimal  as  a  numerator,  and  place  as  the  denominator  1  with  as  many 
ciphers  annexed  as  there  are  decimal  places  in  the  numerator;  erase  the 


ARITHMETIC. 


S3.  $ 


$  2  8 

NO  <s  rx 
t>  00  CO 


i  §  5  3 

>q  t>*  r>»  cq 


in  T  en  —  Q 

eM  ON  NO  en  O 

\O  o  in  o  m 

m  NO  \o  !>•  t> 


t>»  \o  \o  NO  m  m 

rq  m  oo  —  T  i>» 

t>»  «—  m  o  -<r  GO 

•«r  »n  in  \o  NO  vo 

O  ON  oo  fN  >o  «n  in 
en  5r  ^T  S  in  S  2 

^  ^o  t>i  o*  o  R  SS  iS 
vo*—  \O'—  rNtsr-NC^ 
—  moofS>nON<SNO 
c<^  en  en  "^  ^t  ^.  "^  *f\ 

O  en  ir\  06  O  en  iin  oo  O 
o  —  fScn«nspr>.GOO 
moo.  —  •^•t>«OenvOO 

^  oo    —    ^-    06    '—    >n    GO    <s    »n 

—  oovOcnoooineNOrs 

ON  —    •^•rNOCNimcO'—    en 

"1 ;  ts    c^i    CM    en    en    en    en    ^    ^ 

O  —  «n  O^  ^  00  en  t>,  —  \O  O 
O'^-t>.O^r>'-^cO  —  in 
•^•NOoo  —  enmooor4inr>. 

~~.  ^~.  ".  ^  N.  ^  °i  ^  **!  ^  "i 

t>»     <N     t^     (S     00     en     OO     ^     O^     TO     «n 
fNrNNOvOininT-tenenenfN 

§    -.    -.    -.    -.    2    jq    cs    CM    cs    ^   Jn 

u^  —  F>  <n  o  v5  Fi  ON  in  ^  F>  ^  O 
eNQOenONinONO  —  rxenoo^O 
«Ot>«ONO(M'^-inrNOOO'—  enm 
o  O  o  —  —  —  — 4  —  —  cs  cs  cs  CM 

c\i  ^  \o  en  o  oo  in  <s  ON  so  en  *—  oo  «n 
inOoOO«Senint>,oooeN'<r«r>tN, 
enTmt>»ooONO^~eN'^'inNOt>»oo 
OOOOOO  —  —  —  —  •~^1™^^^ 

sO^en    —   ONi>»tnen'—    ONCOO^TP^O 

inen^—  O^NO^<SOoomen^-  ONt>«»n 
^-fNenen-*inNOt>»r>.ooONOO  —  c^ 
o  o  o  o  o  o  o  o  o  o  o  '-;«-;'-;  *~. 

ONOot^^om^fenmvN  —  oONcor%>ovn 
c^J5;^.iXOvcnr>.'-inoNenvOo^oocN 
OO  —  •—  •—  (SeNenenrn^^mininN£ 
OOOOOOOOOOOOOOOC3 


fMoom  —  r>enNN'- 

;j  ~^  «sj  en  en  "t  «r>  in  o.  <q  r>.  cq  cq  O; 


COMPOUND   NUMBERS. 


decimal  point  In  the  numerator,  and  reduce  the  fraction  thus  formed  to  Its 
lowest  terms: 


To  reduce  a  recurring  decimal  to  a  common  fraction.  —  Subtract 
the  decimal  figures  that  do  not  recur  from  the  whole  decimal  including 
one  set  of  recurring  figures;  set  down  the  remainder  as  the  numerator  of 
the  fraction,  and  as  many  nines  as  there  are  recurring  figures,  followed  by 
as  many  ciphers  as  there  are  non-recurring  figures,  in  the  denominator. 
Thus: 

0.79054054,  the  recurring  figures  being  054. 
Subtract    __  79 

7807  'i  117 

99900  "*  (redllced  to  its  l°west  terms)    —  * 


.COMPOUND    OR    DENOMINATE    NUMBERS. 

Reduction  descending. — To  reduce  a  compound  number  to  a  lower 
denomination.  Multiply  the  number  by  as  many  units  of  the  lower 
denomination  as  makes  one  of  the  higher. 


, 

fr 
ae 


3  yards  to  inches:  3  X  36  =  108  inches. 
0.04  square  feet  to  square  inches:  .04  X  144 


•  5.76  sq.  in. 


_  the  given  number  is  in  more  than  one  denomination  proceed  in  steps 
from  the  highest  denomination  to  the  next  lower,  and  so  on  to  the  lowest, 
adding  in  the  units  of  each  denomination  as  the  operation  proceeds. 

3  yds.  1  ft.  7  in.  to  inches:  3X3  =  9,4-1=10,  10  X  12  =  120, +7  =  127 in. 

Reduction  ascending.  —  To  express  a  number  of  a  lower  denomina- 
tion in  terras  of  a  higher,  divide  the  number  by  the  number  of  units  of 
the  lower  denomination  contained  in  one  of  the  next  higher;  the  quotient 
is  in  the  higher  denomination,  and  the  remainder,  if  any,  in  the  lower. 
127  inches  to  higher  denomination. 

127  -^  12  =  10  feet  +  7  inches;  10  feet  •*-  3  =  3  yards  4-  1  foot. 

Ans.  3  yds.  1  ft.  7  in. 

To  express  the  result  in  decimals  of  the  higher  denomination,  divide  the 
given  number  by  the  number  of  units  of  the  given  denomination  contained 
in  one  of  the  required  denomination,  carrying  the  result  to  as  many  places 
of  decimals  as  may  be  desired. 

127  inches  to  yards:     127  -^  36  •=  319/ae  =  3.5277  4-  yards. 

Decimals  of  a  Foot  Equivalent  to  Inches  and  Fractions 
of  an  Inch. 


Inches 

0 

H 

X 

H 

H 

ft 

X 

% 

0 

0 

.01042 

.02083 

.03125 

.04167 

.05208 

.06250 

.07292 

1 

.0833 

.0938 

.1042 

.1146 

.1250 

.1354 

.1458 

.1563 

2 

.1667 

.1771 

.1875 

.1979 

.2083 

.2188 

.2292 

.2396 

3 

.2500 

.2604 

.2708 

.2813 

.2917 

.3021 

.3125 

.3229 

4 

.3333 

.3438 

.3542 

.3646 

.3750 

.3854 

.3958 

.4063 

5 

.4167 

.4271 

.4375 

.4479 

.4583 

.4688 

.4792 

.4896 

.  6 

.5000 

.5104 

.5208 

.5313 

.5417 

.5521 

.5625 

.5729 

7 

.5833 

.5938 

.6042 

.6146 

.6250 

.6354 

.6458 

.6563 

8 

.6667 

.6771 

.6875 

.6979 

.7083 

.7188 

.7292 

.7396 

9 

.7500 

.7604 

.7708 

.7813 

.7917 

.8021 

.8125 

.8229 

10 

.8333 

.8438 

.8542 

.8646 

.8750 

.8854 

.8958 

.9063 

11 

.9167 

.9271 

.9375 

.9479 

.9583 

.9688 

.9792 

.9896 

ARITHMETIC. 


RATIO  AND  PROPORTION. 

Ratio  Is  the  relation  of  one  number  to  another,  as  obtained  by  dividing 
the  first  number  by  the  second.  Synonymous  with  quotient. 

Ratio  of  2  to  4,  or  2  :  4  =  2/4=  l/2. 
Ratio  of  4  to  2,  or  4  :  2  =  2. 

Proportion  is  the  equality  of  two  ratios.  Ratio  of  2  to  4  equals  ratio 
of  3  to  6,  2/4=3/6;  expressed  thus,  2  :  4  ::  3  :  6;  read,  2  is  to  4  as  3  is  to  6. 

The  first  and  fourth  terms  are  called  the  extremes  or  outer  terms,  the 
second  and  third  the  means  or  inner  terms. 

The  product  of  the  means  equals  the  product  of  the  extremes: 

2  :  4  :  :  3  :  6;     2  X  6  =  12;     3  X  4  =  12. 

Hence,  given  the  first  three  terms  to  find  the  fourth,  multiply  the 
second  and  third  terms  together  and  divide  by  the  first. 

2  :  4  : :  3  :  what  number?     Ans.    ~-^  =  6. 

Algebraic  expression  of  proportion.  — a  :  b  :  :  c  :  d;  r  =  -;  ad  *»5c; 

be  be    ,      ad,  ad 

from  which  a  =   -r  ;  d=  —  ;  6=  —  ;   c  =  -7-  • 
a  a  c  o 

From  the  above  equations  may  also  be  derived  the  following: 
6  :  a::d  :  c        a  +  b  :  a  :  :c  +  d  :  c'       a  +  b  :  a  —  b  :  :  c  +  d  ;  c  —  d 
a  :  c  : :  b  :  d        a  +  b  :  b  : :  c  +  d  :  d        an  :  b™  :  _:  cn  :  dn 
a-.b^cid        a  -b:b::c  -  d:d        ^  :   ty  :  :  ^/c  ^ 
a  —  b  :  a:  :c  —  d  :  c 

Mean  proportional  between  two  given  numbers,  1st  and  2d,  is  such 
a  number  that  the  ratio  which  the  first  bears  to  it  equals  the  ratio  which  it 
bears  to  the  second.  Thus,  2:4::4:8;4isa  mean  proportional  between 

2  and  8.     To  find  the  mean  proportional  between  two  numbers,  extract 
the  square  root  of  their  product. 

Mean  proportional  of  2  and  8  =  V2  X  8  =  4. 

Single  Rule  of  Three;  or,  finding  the  fourth  term  of  a  proportion 
when  three  terms  are  given.  —  Rule,  as  above,  when  the  terms  are  stated 
in  their  proper  order,  multiply  the  second  by  the  third  and  divide  by  the 
first.  The  difficulty  is  to  state  the  terms  in  their  proper  order.  The 
term  which  is  of  the  same  kind  as  the  required  or  fourth  term  is  made  the 
third;  the  first  and  second  must  be  like  each  other  in  kind  and  denomina- 
tion. To  determine  which  is  to  be  made  second  and  which  first  requires 
a  little  reasoning.  If  an  inspection  of  the  problem  shows  that  the  answer 
should  be  greater  than  the  third  term,  then  the  greater  of  the  other  two 
given  terms  should  be  made  the  second  term  —  otherwise  the  first.  Thus, 

3  men  remove  54  cubic  feet  of  rock  in  a  day;  how  many  men  will  remove 
in  the  same  time  10  cubic  yards?    The  answer  is  to  be  men  —  make  men 
third  term;  the  answer  is  to  be  more  than  three  men,  therefore  make  the 
greater  quantity,  10  cubic  yards,  the  second  term;  but  as  it  is  not  the  same 
denomination  as  the  other  term  it  must  be  reduced,  =  270  cubic  feet. 
The  proportion  is  then  stated: 

3  X  270 
54  :  270  : :  3  :  x  (the  required  number);  x  =  — ^ir~  =  15  men. 

O'x 

The  problem  is  more  complicated  if  we  increase  the  number  of  given 
terms.  Thus,  in  the  above  question,  substitute  for  the  words  "in  the 
same  time"  the  words  '*  in  3  days."  First  solve  it  as  above,  as  if  the  work 
were  to  be  done  in  the  same  time;  then  make  another  proportion,  stating 
it  thus:  If  15  men  do  it  in  the  same  time,  it  will  take  fewer  men  to  do  it  in 
3  days;  make  1  day  the  second  terra  and  3  days  the  first  term,  3:1:: 
15  men  :  5  men. 


POWERS    OF    NUMBERS. 


. 

FJ 


Compound  Proportion,  or  Double  Rule  of  Three.  —  By  this  rule 
are  solved  questions  like  the  one  just  given,  in  which  two  or  more  statings 
are  required  by  the  single  rule  of  three.  In  it,  as  in  the  single  rule,  there 
is  one  third  term,  which  is  of  the  same  kind  and  denomination  as  the 
fourth  or  required  term,  but  there  may  be  two  or  more  first  and  second 
terms.  Set  down  the  third  term,  take  each  pair  of  terms  of  the  same  kinc1 
separately,  and  arrange  them  as  first  and  second  by  the  same  reasoning  as 
is  adopted  in  the  single  rule  of  three,  making  the  greater  of  the  pair  the 
second  if  this  pair  considered  alone  should  require  the  answer  to  be»greater. 

Set  down  all  the  first  terms  one  under  the  other,  and  likewise  all  the 
second  terms.  Multiply  all  the  first  terms  together  and  all  the  second 
terms  together.  Multiply  the  product  of  all  the  second  terms  by  the  third 
term,  and  divide  this  product  by  the  product  of  all  the  first  terms. 
Example:  If  3  men  remove  4  cubic  yards  in  one  day,  working  12  hours  a 
day,  how  many  men  working  10  hours  a  day  will  remove  20  cubic  yards 
in  3  days? 

Yards  4        90 

: :  3  men  :  x  men . 

Products  120     240  : :  3  :  6  men.     Ans. 

To  abbreviate  by  cancellation,  any  one  of  the  first  terms  may  cancel 
either  the  third  or  any  of  the  second  terms;  thus,  3  in  first  cancels  3  in 
third,  making  it  1,  10  cancels  into  20  making  the  latter  2,  which  into  4 
makes  it  2,  which  into  12  makes  it  6.  and  the  figures  remaining  are  only 
1  :  6  : :  1  :  6. 


Yards 
Days 
Hours 

4 
3 
10 

20 
1 
12 

INVOLUTION,  OR  POWERS  OF  NUMBERS. 


Involution  is  the  continued  multiplication  of  a  number  by  itself  a  given 
number  of  times.  The  number  is  called  the  root,  or  first  power,  and  the 
products  are  called  powers.  The  second  power  is  called  the  square  and 
the  third  power  the  cube.  The  operation  may  be  indicated  without  being 
performed  by  writing  a  small  figure  called  the  index  or  exponent  to  the 
right  of  and  a  little  above  the  root;  thus,  33  =  cube  of  3,  =  27. 

To  multiply  two  or  more  powers  of  the  same  number,  add  their  expo- 
nents; thus,  22  X  23  =  25,  or  4  X  8  =  32  =  25. 

To  divide  two  powers  of  the  same  number,  subtract  their  exponents; 


thus,  23  -*•  22  =  2l  =  2;    22  -s-  24  =  2~2  =.£5 


The  exponent  may 


thus  be  negative.  23  -f-  23  =  2°  =  1,  whence  the  zero  power  of  any 
number  =  1.  The  first  power  of  a  number  is  the  number  itself.  The 
exponent  may  be  fractional,  as  2*,  2$,  which  means  that  the  root  is  to  be 
raised  to  a  power  whose  exponent  is  the  numerator  of  the  fraction,  and 
the  root  whose  sign  is  the  denominator  is  to  be  extracted  (see  Evolution). 
The  exponent  may  be  a  decimal,  as  2°'5,  21'5;  read,  two  to  the  five-tenths 
power,  two  to  the  one  and  five-tenths  power.  These  powers  are  solved  by 
means  of  Logarithms  (which  see). 

First  Nine  Powers  of  the  First  Nine  Numbers. 


^1 

b 

o 

^ 

4th 

5th 

6th 

7th 

8th 

9th 

J§ 

s§ 

en  § 

Power. 

Power. 

Power. 

Power. 

Power. 

Power. 

PL. 

PH 

PH 

1 

, 

1 

1 

1 

1 

1 

1 

1 

2 

4 

8 

16 

32 

64 

128 

256 

512 

3 

9 

27 

81 

243 

729 

2187 

6561 

19683 

A 

16 

64 

256 

1024 

4096 

16384 

65536 

262144 

5 

25 

125 

625 

3125 

15625 

78125 

390625 

1953125 

6 

36 

216 

1296 

7776 

46656 

279936 

1679616 

10077696 

7 

49 

343 

2401 

16807 

1  1  7649 

823543 

5764801 

40353607 

8 

64 

512 

4096 

32768 

262144 

2097152 

16777216 

134217728 

9 

81 

729 

6561 

59049 

531441 

4782969 

43046721 

387420489 

ARITHMETIC, 


The  First  Forty  Powers  of  2. 


0 

I 

o 

Q 

J3 

o 

QJ 

1 

1 

J3 

i 

O 

1 

£ 

> 

^ 

PH 

*" 

ft 

> 

ft 

0 

, 

9 

512 

18 

262144 

27 

134217728 

36 

68719476736 

1 

2 

10 

1024 

19 

524288 

28 

268435456 

37 

137438953472 

2 

4 

11 

2048 

20 

1048576 

29 

536870912 

38 

274877906944 

3 

8 

12 

4096 

21 

2097152 

30 

1073741824 

39 

549755813888 

4 

16 

13 

8192 

22 

4194304 

31 

2147483648 

40 

1099511627776 

5 

32 

14 

16384 

23 

8388608 

32 

4294967296 

6 

64 

15 

32768 

24 

16777216 

33 

8589934592 

7 

128 

16 

65536 

25 

33554432 

34 

17179869184 

8 

256 

17 

131072 

26 

67108864 

35 

34359738368 

EVOLUTION. 

Evolution  is  the  finding  of  the  root  (or  extracting  the  root)  of  any 
number  the  power  of  which  is  given. 

The  sign  V  indicates  that  the  square  root  is  to  be  extracted:  ^  <\J  <^/ 
the  cube  root,  4th  root,  nth  root. 

A  fractional  exponent  with  1  for  the  numerator  of  the  fraction  is  also 
used  to  indicate  that  the  operation  of  extracting  the  root  is  to  be  per- 
formed; thus,  2*,  2*  =  <\/2,  -\/2. 

When  the  power  of  a  number  is  indicated,  the  involution  not  being  per- 
formed, the  extraction  of  any  root  of  that  power  may  also  be  indicated  by 
dividing  the  index  of  the  power  by  the  index  of  the  root,  indicating  the 
division  by  a  fraction.  Thus,  extract  the  square  root  of  the  6th  power 
of  2: 

*/2«  =  2*  =  2*  =  23  =  8. 

The  6th  power  of  2,  as  in  the  table  above,  is  64:  v'ei  =  8. 

Difficult  problems  in  evolution  are  performed  by  logarithms,  but  the 
square  root  and  the  cube  root  may  be  extracted  directly  according  to  the 
rules  given  below.  The  4th  root  is  the  square  root  of  the  square  root. 
The  6th  root  is  the  cube  root  of  the  square  root,  or  the  square  root  of  the 
cube  root;  the  9th  root  is  the  cube  root  of  the  cube  root;  etc. 

To  Extract  the  Square  Root.  —  Point  off  the  given  number  into 
periods  of  two  places  each,  beginning  with  units.  If  there  are  decimals, 
point  these  off  likewise,  beginning  at  the  decimal  point,  and  supplying 
as  many  ciphers  as  may  be  needed.  Find  the  greatest  number  whose 
square  is  less  than  the  first  left-hand  period,  and  place  it  as  the  first 
figure  in  the  quotient.  Subtract  its  square  from  the  left-hand  period, 
and  to  the  remainder  annex  the  two  figures  of  the  second  period  for 
a  dividend.  Double  the  first  figure  of  the  quotient  for  a  partial  divisor; 
find  how  many  times  the  latter  is  contained  in  the  dividend  exclusive 
of  the  right-hand  figure,  and  set  the  figure  representing  that  number  of 
times  as  the  second  figure  in  the  quotient,  and  annex  it  to  the  right  of 
the  partial  divisor,  forming  the  complete  divisor.  Multiply  this  divisor 
by  the  second  figure  in  the  quotient  and  Subtract  the  product  from  the 
dividend.  To  the  remainder  bring  down  the  next  period  and  proceed  as 
before,  in  each  case  doubling  the  figures  in  the  root  already  found  to  obtain 
the  trial  divisor.  Should  the  product  of  the  second  figure  in  the  root  by 
the  completed  divisor  be  greater  than  the  dividend,  erase  the  second 
figure  both  from  the  quotient  and  from  the  divisor,  and  substitute  the 
next  smaller  figure,  or  one  small  enough  to  make  the  product  of  the  second 
figure  by  the  divisor  less  than  or  equal  to  the  dividend. 


6QUA 
o  i  A  1  rcnofl 


CUBE   ROOT. 


SQUARE   ROOT. 

3.1415926536  U/77245  -f 
1 

27(214 
1189 

34712515 
(2429 

354218692 
7084 


CUBE    ROOT. 


35444  160865 
1141776 

55448511908936 
)1772425 


300  X  I2 
30  X  1 


1.881.365.963.6251 12345 
1 

=  300  881 
X2  =    60 
22=      4 

364  728 


300X122  =43200 

30  X  12     X  3    =   1080 

32  =          9 


44289 


I 

300  X  1232  =   4538700 

30  X  123   X  4    =        14760 

42=  16 


4553476 

300X12342        =456826800 

30X1234X5=        185100 

52=  25 


457011925 


20498963 


18213904 


2285059625 


2285059625 


To  extract  the  square  root  of  a  fraction,  extract  the  root  of  a  numerator 

/4~      2 
and  denominator  separately,    1/g  =  ~»  or  first  convert  the  fraction  into 

a  decimal,   *\|  =  V.4444  4-  =  0.6666  -K 

To  Extract  the  Cube  Root.  —  Point  off  the  number  into  periods  of  3 
figures  each,  beginning  at  the  right  hand,  or  unit's  place.  Point  off 
decimals  in  periods  of  3  figures  from  the  decimal  point.  Find  the  greatest 
cube  that  does  not  exceed  the  left-hand  period;  write  its  root  as  the  first 
figure  in  the  required  root.  Subtract  the  cube  from  the  left-hand  period, 
and  to  the  remainder  bring  down  the  next  period  for  a  dividend. 

Square  the  first  figure  of  the  root;  multiply  by  300,  and  divide  the 
product  into  the  dividend  for  a  trial  divisor;  write  the  quotient  after 
the  first  figure  of  the  root  as  a  trial  second  figure. 

Complete  the  divisor  by  adding  to  300  times  the  square  of  the  first 
figure,  30  times  the  product  of  the  first  by  the  second  figure,  and  the 
square  of  the  second  figure.  Multiply  this  divisor  by  the  second  figure; 
subtract  the  product  from  the  remainder.  (Should  the  product  be  greater 
than  the  remainder,  the  last  figure  of  the  root  and  the  complete  divisor 
are  too  large;  substitute  for  the  last  figure  the  next  smaller  number,  and 
correct  the  trial  divisor  accordingly.) 

To  the  remainder  bring  down  the  next  period,  and  proceed  as  before  to 
find  the  third  figure  of  the  root  —  that  is,  square  the  two  figures  of  the 
root  already  found;  multiply  by  300  for  a  trial  divisor,  etc. 

If  at  any  time  the  trial  divisor  is  greater  than  the  dividend,  bring  down 
another  period  of  3  figures,  and  place  0  in  the  root  and  proceed. 

The  cube  root  of  a  number  will  contain  as  many  figures  as  there  are 
periods  of  3  in  the  number. 

To  Extract  a  Higher  Root  than  the  Cube.  —  The  fourth  root  is  the 
square  root  of  the  square  root;  the  sixth  root  is  the  cube  root  of  the  square 
root  or  the  square  root  of  the  cube  root.  Other  roots  are  most  conve- 
niently found  by  the  use  of  logarithms. 

ALLIGATION. 

shows  the  value  of  a  mixture  of  different  ingredients  when  the  quantity 
and  value  of  each  are  known. 

Let  the  ingredients  be  a,  b,  c,  d,  etc.,  and  their  respective  values  per 
unit  w,  x,  y,  z,  etc. 


10  ARITHMETIC. 

A  «=  the  sum  of  the  quantities  =  a+b+c+dt  etc. 
P  =  mean  value  or  price  per  unit  of  A. 
AP  =  aw  +  bx  +  cy  +  dz,  etc. 
P  =  aw  +  bx  +  cy  +  dz 
A 

PERMUTATION 

shows  in  how  many  positions  any  number  of  things  may  be  arranged  in  a 
row;  thus,  the  letters  a,  b,  c  may  be  arranged  in  six  positions,  viz.  abc,  acb, 
cab,  cba,  bac,  bca. 

Rule.  —  Multiply  together  all  the  numbers  used  in  counting  the  things; 
thus,  permutations  of  1,  2,  and  3  =  1X2X3  =  6.  In  how  many 
positions  can  9  things  in  a  row  be  placed? 

1X2X3X4X5X6X7X8X9  =  362880. 

COMBINATION 

shows  how  many  arrangements  of  a  few  things  may  be  made  out  of  a 
greater  number.  Rule:  Set  down  that  figure  which  indicates  the  greater 
number,  and  after  it  a  series  of  figures  diminishing  by  1,  until  as  many  are 
set  down  as  the  number  of  the  few  things  to  be  taken  in  each  combination. 
Then  beginning  under  the  last  one,  set  down  said  number  of  few  things; 
then  going  backward  set  down  a  series  diminishing  by  1  until  arriving 
under  the  first  of  the  upper  numbers.  Multiply  together  all  the  upper 
numbers  to  form  one  product,  and  all  the  lower  numbers  to  form  another; 
divide  the  upper  product  by  the  lower  one. 

How  many  combinations  of  9  things  can  be  made,  taking  3  in  each  com- 
bination? 

9X8X7  _  504  _ 

1X2X3"     6 

ARITHMETICAL  PROGRESSION, 

in  a  series  of  numbers,  is  a  progressive  increase  or  decrease  in  each  succes- 
sive number  by  the  addition  or  subtraction  of  the  same  amount  at  each 
step,  as  1,  2,  3,  4,  5,  etc.,  or  15,  12,  9,  6,  etc.  The  numbers  are  called  terms, 
and  the  equal  increase  or  decrease  the  difference.  Examples  in  arithmeti- 
cal progression  may  be  solved  by  the  following  formulae: 

Let  a  =  first  term,  I  =  last  term,  d  =  common  difference,  n  =  number 

of  terms,  s  =  sum  of  the  terms; 

1  /  /         1    \2 

I  =  a  +  (n  —  l)d,  =  —  -  d  ±  y  2ds  -f  I  a  —  -  d\  9 

2s  s    ,    (n  —  I)d 

~  ~n  ~~  a>  =  ri        — 2 — 


X"*J*  2  2d 

2  2 


==id±  Id  4-ldV-- 


l-a 
d-^-l* 

P  -  a 


'  2s  -  I  —  a 

I  -  a    , 

~T~   *"  *• 

2s 
!  I  +  a '  2d 


2(s  -  an) 

n(n  -  1)  ' 
2(nl  -  s) 

n(n  -  1) 

d  —  2a  ±  V(2a  -  < 

*)2  +  8ds 

2d 

21  +  d  ±  ^(21  -f  d)! 

*  -  Sds 

GEOMETRICAL   PROGRESSION. 


GEOMETRICAL  PROGRESSION. 


11 


»ix  ci  series  of  numbers,  is  a  progressive  increase  or  decrease  in  each  suc- 
cessive number  by  the  same  multiplier  or  divisor  at  each  step,  as  1,  2,  4,  8, 
16,  etc.,  or  243,  81,  27,  9,  etc.  The  common  multiplier  is  called  the  ratio. 
Let  a  =  first  term,  I  =  last  term,  r  =  ratio  or  constant  multiplier,  n  = 
number  of  terms,  m  =  any  term,  as  1st,  2d,  etc.,  s  =  sum  of  the  terms: 


log  I  =  log  a  +  (n  -  1)  logr,  l(s  —  J)71""1  -  a(s  -  a)n~l  =  0, 

m  =  arm—1  log  m  =  log  a  +  (m  —  1)  log  r. 

n~^-n^/a^       _     lrn_l 


rl-a 


log?  -  log  a 

logr            h1' 
log  I  —  log  a 

~~  log  (s  —  a)  —  log  (s  -  I) 


log  [a  +  (r  -  l)s]  -  log  a  ( 

logr 
log?  —  log  [Ir  —  (r  —  l)s] 

logr 


Population  of  the  United  States. 

(A  problem  in  geometrical  progression.) 


Year. 

1860 
1870 
1880 
1890 
1900 
1910 
1920 


Population. 

31,443,321 
39,818,449* 
50,155,783 
62,622,250 
76,295,220 
91,972,267 
Est.  110,367,000 


Increase  in  10    Annual  Increase, 
Years,  per  cent.          per  cent. 


26.63 
25.96 
24.86 
21.834 
20.53 
Est.  20.0 


2.39 
2.33 
2.25 
1.994 
1.886 
Est.  1.840 


Estimated  Population  in  Each  Year  from  1880  to  1919. 
(Based  on  the  above  rates  of  increase,  in  even  thousands.) 


I860. 

50,156 

1890. 

62.622 

1900. 

76.295 

1910.. 

91.972 

1881. 

51,281 

1891. 

63.871 

1901  . 

77.734 

1911  .. 

93.665 

1882. 

52.433 

1892. 

65.145 

1902. 

79.201 

1912.. 

95.388 

1883. 

53.610 

1893. 

66444 

1903. 

80.695 

1913.. 

97,143 

1884. 

54.813 

1894. 

67.770 

1904. 

82.217 

1914.. 

98.930 

1885. 

56,043 

1895. 

69,122 

1905. 

83.768 

1915.. 

100.750 

1886. 

57.301 

1896. 

70.500 

1906. 

85.348 

1916.. 

102.604 

1887. 

58,588 

1897. 

71.906 

1907. 

86.958 

1917.. 

104.492 

1888. 

59.903 

1898. 

73.341 

1908. 

88.598 

1918.. 

106.414 

1889. 

61,247 

1899. 

74.803 

1909. 

90.269 

1919.. 

108.373 

*  Corrected  by  addition  of  1,260,078,  estimated  error  of  the  census  of 
1870,  Census  Bulletin  No.  16,  Dec,  13, 1890. 


12  ARITHMETIC. 

The  preceding  table  has  been  calculated  by  logarithms  as  follows: 
log  r  =  log  I  —  log  a  -5-  (n  —  1),  log  m  =  log  a  +  (m  -  1)  log  f 

Pop.  1900.  .  .76,295,220  log  =  7.8824988  =  log  I 

1890.  .  .62,622,250  log  =  7.7967285  =  log  a 

diff.  =     .0857703 

n  «=  11,  n  -  1  =  10;  diff.  -J-  10  =     .00857703          =  log  r, 
add  log  for  1890       7.7967285  •=  log  a 

log  for  1891  =  7.80530553  No.  =  63,871  . . 
add  again         .00857703 

log  for  1892       7.81388256  No.  =  65,145  . . . 

Compound  interest  is  a  form  of  geometrical  progression;  the  ratio 
being  1  plus  the  percentage. 


PERCENTAGE:  PROFIT  AND  LOSS:    PER  CENT 
OF  EFFICIENCY. 

Per  cent  means  "by  the  hundred."  A  profit  of  10  per  cent  means  a 
gain  of  $10  on  every  $100  expended.  If  a  thing  is  bought  for  $1  and  sold 
for  $2  the  profit  is  100  per  cent;  but  if  it  is  bought  for  $2  and  sold  for  $1 
the  loss  is  not  100  per  cent,  but  only  50  per  cent. 

Rule  for  percentage:  Per  cent  gain  or  loss  is  the  gain  or  loss  divided  by 
the  original  cost,  and  the  quotient  multiplied  by  100. 

Efficiency  is  defined  in  engineering  as  the  quotient  "output  divided  by 
input,"  that  is,  the  energy  utilized  divided  by  the  energy  expended.  The 
difference  between  the  input  and  the  output  is  the  loss  or  waste  of  energy. 
Expressed  as  a  fraction,  efficiency  is  nearly  always  less  than  unity.  Ex- 
pressed as  a  per  cent,  it  is  this  fraction  multiplied  by  100.  Thus  we  may 
say  that  a  motor  has  an  efficiency  of  0.9  or  of  90  per  cent. 

The  efficiency  of  a  boiler  is  the  ratio  of  the  heat  units  absorbed  by  the 
boiler  in  heating  water  and  making  steam  to  the  heating  value  of  the  coal 
burned.  The  saving  in  fuel  due  to  increasing  the  efficiency  of  a  boiler 
from  60  to  75%  is  not  25%,  but  only  20%.  The  rule  is:  Divide  the  gain 
in  efficiency  (15)  by  the  greater  figure  (75).  The  amount  of  fuel  used  is 
inversely  proportional  to  the  efficiency;  that  is,  60  Ibs.  of  fuel  with  75% 
efficiency  will  do  as  much  work  as  75  Ibs.  with  60%  efficiency.  The 
saving  of  fuel  is  15  lb*.  which  is  20%  of  75  Ibs. 


INTEREST  AND  DISCOUNT. 

Interest  is  money  paid  for  the  use  of  money  for  a  given  time;  tho 
factors  are: 

p,  the  sum  loaned,  or  the  principal; 

t,  the  time  in  years; 

r,  the  rate  of  interest ; 

i,  the  amount  of  interest  for  the  given  rate  and  time; 

a  =  p  +  i  =    the  amount  of  the  principal  with  interest 

at  the  end  of  the  time. 
Formulas: 

i  —  interest  =  principal  X  time  X  rate  per  cent  =  i  =  J-QQ  I 

a  —  amount  =  principal  +  interest  =  p  +  ^g  •' 
lOOi 


r  -rate- 


INTEREST  AND   DISCOUNT.  33 

If  the  rate  is  expressed  decimally,  —  thus,  6  per  cent  =  .06, —  the 
formulse  become 

Rules  for  finding  Interest.  —  Multiply  the  principal  by  the  rate  per 
annum  divided  by  100,  and  by  the  time  in  years  and  fractions  of  a  year. 

If  the  time  is  given  in  days,  interest  =  Principal  X  rate  X  no.  of  Jays  _ 

ooo  X  100 

In  banks  interest  is  sometimes  calculated  on  the  basis  of  360  days  to  a 
year,  or  12  months  of  30  days  each. 

Short  rules  for  interest  at  6  per  cent,  when  360  days  are  taken  as  1  year: 

Multiply  the  principal  by  number  of  days  and  divide  by  6000. 

Multiply  the  principal  by  number  of  months  and  divide  by  200. 

The  interest  of  1  dollar  for  one  month  is  £  cent. 

Interest  of  10O  Dollars  for  Different  Times  and  Rates. 

Time  3%         3%        4%         5%         6%         8%         10% 

lyear  $2.00      $3.00      $4.00      $5.00      $6.00       $8.00     $10.00 

1  month  .16|         .25          .33£         .41§        .50  .66|         .83$ 

lday=g|5year.0055i    .0083£    .0111$     .0138f     .0166§    .0222§      .02775 

is  year  .005479  .008219  .010959  .013699  .016438  .0219178  .0273973 


Discount  is  interest  deducted  for  payment  of  money  before  it  is  due. 

True  discount  is  the  difference  between  the  amount  of  a  debt  payable 
at  a  future  date  without  interest  and  its  present  worth.  The  present 
worth  is  that  sum  which  put  at  interest  at  the  legal  rate  will  amount  to 
the  debt  when  it  is  due. 

To  find  the  present  worth  of  an  amount  due  at  a  future  date,  divide  the 
amount  by  the  amount  of  $1  placed  at  interest  for  the  given  time.  The 
discount  equals  the  amount  minus  the  present  worth. 

What  discount  should  be  allowed  on  $103  paid  six  months  before  it  is 
due,  interest  being  6  per  cent  per  annum? 

— ?  =  $100  present  worth,  discount  =  3.00. 
1  +1  X  .06  X  ^ 

Bank  discount  is  the  amount  deducted  by  a  bank  as  interest  on  money 
loaned  on  promissory  notes.  It  is  interest  calculated  not  on  the  actual 
sum  loaned,  but  on  the  gross  amount  of  the  note,  from  which  the  discount 
is  deducted  in  advance.  It  is  also  calculated  on  the  basis  of  360  days 
in  the  year,  and  for  3  (in  some  banks  4)  days  more  than  the  time  specified 
in  the  note.  These  are  called  days  of  grace,  and  the  note  is  not  payable 
till  the  last  of  these  days.  In  some  States  days  of  grace  have  been 
abolished. 

What  discount  will  be  deducted  by  a  bank  in  discounting  a  note  for  $103 
payable  6  months  hence?  Six  months  =  182  days,  add  3  days  grace  =  185 


Compound  Interest.  —  In  compound  interest  the  interest  is  added  to 
the  principal  at  the  end  of  each  year,  (or  shorter  period  if  agreed  upon). 

Let  p  =  the  principal,  r  =  the  rate  expressed  decimally,  n  =  no.  of 
years,  and  a  the  amount: 


o  —  amount  —  p(l  +  r)n;  r  —  rate  =»   u  -  -  1. 
p  —  principal  =»    (l  £  .n  ;  no.  of  y.ears=-  n  = 


14 


ARITHMETIC. 


Compound  Interest  Table. 

(Value  of  one  dollar  at  compound  interest,  compounded  yearly,  at 
3,  4,  5,  and  6  per  cent,  from  1  to  50  years.) 


£ 

Per  cent 

t 

§ 
** 

Per  cent 

3 

4 

5 

6 

3 

4 

5 

6 

i 

.03 

.04 

.05 

.06 

16 

.6047 

1  .8730 

2.1829 

2.5403 

2 

.0609 

.0816 

.1025 

.1236 

17 

.6528 

1.9479 

2.2920 

2.6928 

3 

.0927 

.1249 

.1576 

.1910 

18 

.7024 

2.0258 

2.4066 

2.8543 

4 

.1255 

.1699 

.2155 

.2625 

19 

.7535 

2.1068 

2.5269 

3.0256 

5 

.1593 

.2166 

.2763 

.3382 

20 

.8061 

2.1911 

2.6533 

3.2071 

6 

.1941 

.2653 

.3401 

.4185 

21 

.8603 

2.2787 

2.7859 

3.3995 

7 

.2299 

.3159 

.4071 

.5036 

22 

.9161 

2.3699 

2.9252 

3.6035 

8 

.2668 

.3686 

.4774 

.5938 

23 

.9736 

2.4647 

3.0715 

3.8197 

9 

.3048 

.4233 

.5513 

.6895 

24 

2.0328 

2.5633 

3.2251 

4.0487 

10 

.3439 

.4802 

.6289 

.7908 

25 

2.0937 

2.6658 

3.3863 

4.2919 

11 

.3842 

.5394 

.7103 

1.8983 

30 

2.4272 

3.2433 

4.3219 

5.7435 

12 

.4258 

.6010 

.7958 

2.0122 

35 

2.8138 

3.9460 

5.5159 

7.6862 

13 

.4685 

.6651 

.8856 

2.1329 

40 

3.2620 

4.8009 

7.0398 

10.2858 

14 

1.5126 

.7317 

.9799 

2.2609 

45 

3.7815 

5.8410 

8.9847 

13.7648 

15 

1.5580 

.8009 

2.0789 

2.3965 

50 

4.3838 

7.1064 

11.4670 

18.4204 

At  compound  interest  at  3  per  cent  money  will  double  itself  in  23 1/2  years, 
at  4  per  cent  in  172/3  years,  at  5  per  cent  in  14.2  years,  and  at  6  per  cent  io 
11. 9  years. 

EQUATION  OF  PAYMENTS. 

By  equation  of  payments  we  find  the  equivalent  or  average  time  in 
which  one  payment  should  be  made  to  cancel  a  number  of  obligations  due 
at  different  dates;  also  the  number  of  days  upon  which  to  calculate  interest 
or  discount  upon  a  gross  sum  which  is  composed  of  several  smaller  sums 
payable  at  different  dates. 

Rule.  —  Multiply  each  item  by  the  time  of  its  maturity  in  days  from  a 
fixed  date,  taken  as  a  standard,  and  divide  the  sum  of  the  products  by 
the  sum  of  the  items:  the  result  is  the  average  time  in  days  from  the  stand- 
ard date. 

A  owes  B  $100  due  in  30  days,  $200  due  in  60  days,  and  $300  due  in  90 
days.  In  how  many  days  may  the  whole  be  paid  in  one  sum  of  $600? 

100X30+200X60+300X90  =  42,000;     42,000-^600  =  70  days,     ans. 

A  owes  B  $100,  $200,  and  $300,  which  amounts  are  overdue  respectively 
30,  60,  and  90  days.  If  he  now  pays  the  whole  amount,  $600,  how  many 
days'  interest  should  he  pay  on  that  sum?  Ans.  70  days. 


PARTIAL,  PAYMENTS. 

To  compute  interest  on  notes  and  bonds  when  partial  payments  have 
been  made. 

United  States  Rule.  —  Find  the  amount  of  the  principal  to  the  time 
of  the  first  payment,  and,  subtracting  the  payment  from  it,  find  the 
amount  of  the  remainder  as  a  new  principal  to  the  time  of  the  next  pay* 
meat. 


ANNUITIES. 


15 


If  the  payment  is  less  than  the  interest,  find  the  amount  of  the  principal 
to  the  time  when  the  sum  of  the  payments  equals  or  exceeds  the  interest 
due,  and  subtract  the  sum  of  the  payments  from  this  amount. 

Proceed  in  this  manner  till  the  time  of  settlement. 

Note.  —  The  principles  upon  which  the  preceding  rule  is  founded  are: 

1st.  That  payments  must  be  applied  first  to  discharge  accrued  interest, 
and  then  the  remainder,  if  any,  toward  the  discharge  of  the  principal. 

2d.   That  only  unpaid  principal  can  draw  interest. 

Mercantile  Method.  —  When  partial  payments  are  made  on  short 
notes  or  interest  accounts,  business  men  commonly  employ  the  following 
method: 

Find  the  amount  of  the  whole  debt  to  the  time  of  settlement ;  also  find 
the  amount  of  each  payment  from  the  time  it  was  made  to  the  time  of 
settlement.  Subtract  the  amount  of  payments  from  the  amount  of  the 
debt:  the  remainder  will  be  the  balance  due. 


ANNUITIES. 

An  Annuity  is  a  fixed  sum  of  money  paid  yearly,  or  at  other  equ^l  times 
agreed  upon.  The  values  of  annuities  are  calculated  by  the  principles  of 
compound  interest. 

1.  Let  i  denote  interest  on  $  1  for  a  year,  then  at  the  end  of  a  year  trier 
amount  will  be  1  +  i.     At  the  end  of  n  years  it  will  be  (1  -f  i)n. 

2.  The  sum  which  in  n  years  will  amount  to  1  is  or  (1  +  i)  —  nf 


or  the  present  value  of  1  due  in  n  years. 

3.  The  amount  of  an  annuity  of  1  in  any  number  of  years  n  is      '     : — —  • 

4.  The  present  value  of  an  annuity  of  1  for  any  number  of  years  n  is 


5.   The  annuity  which  1  will  purchase  for  any  number  of  years  n  la 

i 


6.   The  annuity  which  would  amount  to  1  in  n  years  is  • 


(1  +  i)n  -  , 
Amounts,  Present  Values,  etc.,  at  5%  Interest. 


(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

Years 

CH-t)« 

(l-fzTn 

(l+i)n_i 

1-0+0-" 

i 

i 

I 

i 

1-0  +  0-n 

(1+t-)n_* 

1  . 

.05 

.952381 

1.00 

.952381 

1.05 

1.00 

2. 

.1025 

.907029  ' 

2.05 

1.859410 

.537805 

.487805 

3. 

.157625 

.863838 

3.1525 

2.723248 

.367209 

.317209 

4. 

.215506 

.822702 

4.310125 

3.545951 

.282012 

.232012 

5. 

.276282 

.783526 

5.525631 

4.329477 

.230975 

.180975 

6. 

.340096 

.746215 

6.801913 

5.075692 

.197017 

.147018 

7. 

.407100 

.710681 

8.142008 

5.786373 

.172820 

.122820 

8. 

.477455 

.676839 

9.549109 

6.463213 

.154722 

.104722 

9. 

.551328 

.644609 

1  1  .026564 

7.107822 

.  1  40690 

.090690 

10. 

.628895 

.613913 

12.577893 

7.721735 

.129505 

.079505 

16 


ARITHMETIC. 


J£  "fr  cq  IN.  en      Ov— 'i^iinNO*      o^  <N  IN.'  eN  od     inrsic^ixad     fN  od  NO  "T  en 
GO  — fNixTt-     —  QGOIN.VO     mm  T  T  en     tnencNCN—     — 


tNenOsenoO  vONOenrxhs  enaOaO(SGO  rj-fNinoOm  OrxtNenNO 

NONOeN— ;—  o^  GO  GO  NO  in  ONoeNNOin  oo^  —  NO  in  cqo^fn'<t;o 

NOmoo^in  o'eNoot>.Qo"  — '•^•O^TO  !>.' en  — '  od  O^*  en' O*  t>»  in  ^ 

SzrJTJ^^"  f^ocoi^O  NOin-tn-T  enenencN—  — w 


. 

o  rs  GO  —  O      ^  ts  [  "f  ao 
''  '' 


o  r>.  -< 

[  "f  ao  "1 
'' 


tn  in-^-i^Tr^  OO-^-oo—  c^G 

s  —  tr»  —  \q  >o\  oo  ts  o^  —  m  O  oq 

K-*   vOO^'o'in   csl  ad  >O  <n  •*  hs' 
- 


O*  •—  NO'  m*  —      fx  o^'  in  T  m      ix  —  in  o  NO" 
Tents  —  ~     — —     ^^  «nmT 


t^mO 

—  oo  O 

T-—  oor>» 


TencN  —  —      — —          ^  "^"^ 


p i  fx  —  T  NO     «n  T  -f  csj  o     f  O  m  r> ,  NO     o^  t>.  oq  CN  m     —  IN.  in  —  eN 


NO  m  o  en  vq  In  -^  -^r  f 
''        ' 


)  O      ^r  o  in  ix  NO      O^  r>  ;  oq  ?N  -T      o  «n  csi  r>.  oq 
'  '  '  '' 


co  ONOm^o  txTcntN.o  n-ooooom 

f»«O  in-«t-^-fNiO  •«roini>.Nq  C^.'O.***"^*** 

'in  •—  "m'o^'ooo*  •—  in'o^^'o  vom'ooooo' 

om  «n  —  ONOOI>»  t>  NO  m  m  m  T  ^  ^r  en  CN 


rn'inoONo'  rin-'oao 
o<vj^rcMn  m  —  ooooo 
•^rm«N  —  •—  •-  —  — 


. 

—     "»•  o  in  r>.  NOt  a^  NO  r>  —  r^ 

'oin—  r>.'"r—  '•&<* 

NOmm  T  T  T  en  cq 


V  in  —  —  '  t>C     en  in  —  o—     tntx  —  NO'CN     co'intNO 
OfN^om     m*-oONOO     txvONOinm     T  T  T  "t 


>•  *^  rsi     %o  O  in  o^  o 
O'  en  —  —      ^t"  O  NO'  en  « 

rT^cn     CSCN  —  — 


IH  o 


WEIGHTS   AND    MEASURES.  17 


TABLES  FOB  CALCULATING  SINKING-FUNDS  AND 
PRESENT  VALUES. 

Engineers  and  others  connected  with  municipal  work  and  industrial 
enterprises  often  find  it  necessary  to  calculate  payments  to  sinking-funds 
which  will  provide  a  sum  of  money  sufficient  to  pay  off  a  bond  issue  or 
other  debt  at  the  end  of  a  given  period,  or  to  determine  the  present  value 
of  certain  annual  charges.  The  accompanying  tables  were  computed  by 
Mr.  John  W.  Hill,  of  Cincinnati,  Eng'g  News,  Jan.  25,  1894. 

Table  I  (opposite  page)  shows  the  annual  sum  at  various  rates  of  interest 
required  to  net  $1000  in  from  2  to  50  years,  and  Table  II  shows  the  present 
value  at  various  rates  of  interest  of  an  annual  charge  of  $1000  for  from  5 
to  50  years,  at  five-year  intervals,  and  for  100  years. 

Table  II.  —  Capitalization  of  Annuity  of  $1000  for 
from  5  to  10O  Years. 


1 

Rate  of  IL  terest,  per  cent. 

5 
10 
15 
20 
25 

30 
35 
40 
45 
50 
100 

31/2 

3 

3V2 

4 

4V2 

5 

5V2 

6 

4,645.88 
8,752.17 
12,381.41 
15,589.215 
18,424.67 

20,930.59 
23,145.31 
25,103.53 
26,833.15 
28,362.48 
36,614.21 

4.579.60 
8,530.13 
11,937.80 

4,5  1  4.  92 
8,316.45 
11,517.23 

4,451.68 
8,110.74 
11,118.06 

4,389.91 
7,912.67 
10,739.42 

4,329.45 
7,721.73 
10,379.53 

4,268.09 
7,537.54 
10,037.48 

4,212.40 
7,360.19 
9,712.30 

14,877.27 
17,413.01 

19,600.21 

14,212.12 
16,481.28 

18,391.85 

13,590.21 
15,621.93 

17,291.86 

13,007.88 
14,828.12 

16,288.77 

12,462.13 
14,093.86 

15,372.36 

11,950.26 
13,413.82 

14,533.63 

11,469.96 
12,783.38 

13,764.85 

21,487.04 
23,114.36 
24,518.49 
25,729.58 
31,598.81 

20,000.43 
21,354.83 
22,495.23 
23,455.21 
27,655.36 

18,664.37 
19,792.65 
20,719.89 
21,482.08 
24,504.96 

17,460.89 
18,401.49 
19,156.24 
19,761.93 
21,949.21 

16,374.36 
17,159.01 
17,773.99 
18,255.86 
19,847.90 

15,390.48 
16,044.92 
16,547.65 
16,931.97 
18,095.83 

14,488.65 
15,046.31 
15,455.85 
15,761.87 
16,612.64 

WEIGHTS  AND  MEASURES. 

Long  Measure.  —  Measures  of  Length. 

12  inches  =  1  foot. 

3  feet  =  1  yard. 

1760  yards,  or  5280  feet  =  1  mile. 

Additional  measures  of  length  in  occasional  use:  1000  mils  =  1  inch; 
4  inches  =  1  hand;  9  inches  =  1  span;  2  1/2  feet  =  1  military  pace;  2  yards 
=  1  fathom;  5  1/2  yards,  or  161/2  feet  =  1  rod  (formerly  also  called  pole  or 
perch). 

Old  Land  Measure.  —  7.92  inches  =  1  link;  100  links,  or  66  feet,  or  4 
rods  =  1  chain;  10  chains,  or  220  yards  =  1  furlong;  8  furlongs,  or  80 
chains  =  1  mile;  10  square  chains  =  1  acre. 

Nautical  Measure. 

6080.26JeeU.or  1.15156  stat-  J  =1  nautical 

3  nautical  miles  =1  league. 

60  nautical  miles,  or  69.168  )  _ 
statute  miles  J  - 


/nt  thp  pmiatnr^ 
lat  tne  equator). 


360  degrees 


circumference  of  the  earth  at  the  equator. 


*  The  British  Admiralty  takes  the  round  figure  of  6080  ft.  which  is  the 
length  of  the  "  measured  mile"  used  in  trials  of  vessels.  The  value  varies 
from  6080.26  to  6088.44  ft.  according  to  different  measures  of  the  earth's 
diameter.  There  is  a  difference  of  opinion  among  writers  as  to  the  use 
of  the  word  "  knot"  to  mean  length  or  a  distance  —  some  holding  that 
it  should  be  used  only  to  denote  a  rate  of  speed.  The  length  between 
knots  on  the  log  line  is  1/120  of  a  nautical  mile,  or  50.7  ft.,  when  a  half- 
minute  glass  is  used;  so  that  a  speed  of  10  knots  is  equal  to  10  nautical 
miles  per  hour. 


18  ARITHMETIC. 

Square  Measure.  —  Measures  of  Surface. 

144  square  inches,  or  183.35  circular  )  _  ,  f     . 

inches  )or» 

9  square  feet  =  1  square  yard. 

30V4  square  yards,  or  2721/4  square  feet         ••=  1  square  rod. 
10  sq.  chains,  or  160  sq.  rods,  or  4840  sq.    )       , 

yards,  or  43560  sq.  feet 

640  acres  or  27,878,400  sq.  ft.  =1  square  mile. 

An  acre  equals  a  square  whose  side  is  208.71  feet. 
Circular  Inch;  Circular  Mil.  —  A  circular  inch  is  the  area  of  a  circle 

1  inch  m  diameter  =  0.7854  square  inch. 

1  square  inch  =  1.2732  circular  inches. 

A  circular  mil  is  the  area  of  a  circle  1  mil,  or  0.001  inch  in  diameter. 
10002  or  1,000,000  circular  mils  =-  1  circular  inch. 

1  square  inch  =  1,273,239  circular  mils. 

t   The  mil  and  circular  mil  are  used  in  electrical  calculations  involving 
tne  diameter  and  area  of  wires. 

Solid  or  Cubic  Measure.  —  Measures  of  Volume. 

1728  cubic  inches  =  1  cubic  foot. 
27  cubic  feet      =  1  cubic  yard. 

1  cord  of  wood  =  a  pile,  4X4X8  feet  =  128  cubic  feet. 
1  perch  of  masonry  =  161/2  X  11/2  X  1  foot   =  243/4  cubic  feet. 

Liquid  Measure. 

4  pills      =  1  pint. 
2  pints     =  1  quart. 
4  nnart«    —  i  p-niirm  J  U.  S.  231  cubic  inches. 

-  1  gallon  jEng  277.274  cubic  inches. 

Old  Liquid  Measures.  —  31 1/2  gallons  =  1  barrel;  42  gallons  =  1  tierce; 

2  barrels,  or  63  gallons  =  1  hogshead;  84  gallons,  or  2  tierces  =  1  pun- 
cheon; 2  hogsheads,  or  126  gallons  =  1  pipe  or  butt;  2  pipes,  or  3  pun- 
cheons =  1  tun. 

A  gallon  of  water  at  62°  F.  weighs  8.33531b.  (air  free,  weighed  in  vacuo). 

The  U.  S.  gallon  contains  231  cubic  inches;  7.4805  gallons  =  1  cubic 
foot.  A  cylinder  7  in.  diam.  and  6  in.  high  contains  1  gallon,  very  nearly, 
or  230.9  cubic  inches.  The  British  Imperial  gallon  contains  277.274  cubic 
inches  =  1.20032  U.  S.  gallon,  or  10  ibs.  of  water  at  62°  F. 

The  gallon  is  a  very  troublesome  unit  for  engineers.  Much  labor  might 
be  saved  if  it  were  abandoned  and  the  cubic  fo9t  used  instead.  The 
capacity  of  a  tank  or  reservoir  should.be  stated  in  cubic  feet,  and  the 
delivery  of  a  pump  in  cubic  feet  per  second  or  in  millions  of  cubic  feet  in 
24  hours.  One  cubic  foot  per  second  =  86,400  cu.  ft.  in  24  hours.  One 
million  cu.  ft.  per  24  hours  =  11.5741  cu.  ft.  per  sec. 

The  Miner's  Inch.  —  (Western  U.  S.  for  measuring  flow  of  a  stream 
of  water.)  An  act  of  the  California  legislature,  May  23,  1901,  makes  the 
standard  miner's  inch  1.5  cu.  ft.  per  minute,  measured  through  any  aper- 
ture or  orifice. 

The  term  Miner's  Inch  is  more  or  less  indefinite,  for  the  reason  that  Cali- 
fornia water  companies  do  not  all  use  the  same  head  above  the  centre  of 
the  aperture,  and  the  inch  varies  from  1.36  to  1.73  cu.  ft.  per  min.,  but 
the  most  common  measurement  is  through  an  aperture  2  ins.  high  and 
whatever  length  is  required,  and  through  a  plank  11/4  ins.  thick.  The 
lower  edge  of  the  aperture  should  be  2  ins.  above  the  bottom  of  the  meas- 
uring-box, and  the  plank  5  ins.  high  above  the  aperture,  thus  making  a  6-in. 
head  above  the  centre  of  the  stream.  Each  square  inch  of  this  opening 
represents  a  miner's  inch,  which  is  equal  to  a  flow  of  1 1/2  cu.  ft.  per  min. 

Apothecaries'  Fluid  Measure. 

60  minims  =  1  fluid  drachm.  8  drachms  =  1  fluid  ounce. 

In  the  U.  S.  a  fluid  ounce  is  the  128th  part  of  a  U.  S.  gallon,  or  1.805 
cu.  ins.  It  contains  456.3  grains  of  water  at  39°  F.  In  Great  Britain 
the  fluid  ounce  is  1.732  cu.  ins.  and  contains  1  ounce  avoirdupois,  or  437.5 
grains  of  water  at  62°  F. 


WEIGHTS   AND   MEASURES.  19 

Dry  Measure,  U.  S. 

2  pints  =  1  quart.        8  quarts  =  1  peck.  4  pecks  =  1  bushel. 

The  standard  U.  S.  bushel  is  the  Winchester  bushel,  which  is,  in 
cylinder  form,  18  1/2  inches  diameter  and  8  inches  deep,  and  contains 
2150.42  cubic  inches. 

A  struck  bushel  contains  2150.42  cubic  inches  =  1.2445  cu.  ft.;  1 
cubic  foot  =  0.80356  struck  bushel.  A  heaped  bushel  is  a  cylinder  18 1/2 
inches  diameter  and  8  inches  deep,  with  a  heaped  cone  not  less  than 
6  inches  high.  It  is  equal  to  1 V*  struck  bushels.  (When  applied  to 
apples  and  pears  the  bushel  should  be  heaped  so  as  to  contain  2737.715 
cu.  in.  =  1.2731  struck  bushels. — Decision  of  U.  S.  Court  of  Customs 
Appeals,  1912.) 

The  British  Imperial  bushel  =  8  imperial  gallons  or  2218.192  cu.  in.  = 
1.2837  cu.  ft.  The  British  quarter  =  8  imperial  bushels. 

Capacity  of  a  cylinder  in  U.  S.  gallons  =  square  of  diameter,  in  inches 
X  height  in  inches  X  .0034.  (Accurate  within  1  part  in  100,000.) 

Capacity  of  a  cylinder  in  U.  S.  bushels  =  square  of  diameter  in  inches 
X  height  in  inches  X  0.0003652. 

Shipping  Measure. 

Register  Ton.— For  register  tonnage  or  for  measurement  of  the  entire 
ternal  capacity  of  a  vessel: 

100  cubic  feet  =  1  register  ton. 

This  number  is  arbitrarily  assumed  to  facilitate  computation. 
Shipping  Ton. — For  the  measurement  of  cargo: 

40  cubic  feet  =  1  U.  S.  shipping  ton  =  32.143  U.  S.  bushels. 

42  cubic  feet  =  1  .British  shipping  ton  =  32.719  imperial  bushels. 

Carpenter's  Rule. — Weight  a  vessel  will  carry  =  length  of  keel  X 
breadth  at  main  beam  X  depth  of  hold  in  feet  -h  95  (the  cubic  feet 
allowed  for  a  ton).  The  result  will  be  the  tonnage.  For  a  double- 
decker  instead  of  the  depth  of  the  hold  take  half  the  breadth  of  the 


Measures  of  Weight.— Avoirdupois  or  Commercial 
Weight. 

16  drachms,  or  437.5  grains  =  1  ounce,  oz. 
16  ounces,  or  7000  grains  =  1  pound,  Ib. 
28  pounds  =  1  quarter,  qr. 

4  quarters  =  1  hundredweight,  cwt.  =  112  Ib. 

20  hundredweight  =  1  ton  of  2240  Ib.,  gross  or  long  ton. 

2000  pounds  =  1  net,  or  short  ton. 

2204.6  pounds  =  1  metric  ton. 

1  stone  =  14  pounds;  1  quintal  =  100  pounds. 

The  drachm,  quarter,  hundredweight,  stone,  and  quintal  are  now 
seldom  used  in  the  United  States. 

Troy  Weight 

24  grains  =  1  pennyweight,  dwt. 

20  pennyweights  =  1  ounce,  oz.  =  480  grains. 

12  ounces  =  1  pound,  Ib.  =  5760  grains. 

Troy  weight  is  used  for  weighting  gold  and  silver.  The  grain  is  the 
same  in  Avoirdupois.  Troy,  and  Apothecaries'  weights.  A  carat,  for 
weighing  diamonds  =  3.086  grains  =  0.200  gramme.  (International 
Standard,  1913.) 

Apothecaries'  Weight. 

20  grains      =  1  scruple,  3 

3  scruples  —  1  drachm,  3  -      60  grains. 

8  drachms  »  1  ounce,  5  —    480  grains. 

12  ounces     «  1  pound,  Ib.  «  5760  grains. 


20  ARITHMETIC. 

To  determine  whether  a  balance  has  unequal  arms.  —  After  weigh- 
ing an  article  and  obtaining  equilibrium,  transpose  the  article  and  the 
weights.  If  the  balance  is  true,  it  will  remain  in  equilibrium;  if  untrue, 
the  pan  suspended  from  the  longer  arm  will  descend. 

To  weigh  correctly  on  an  incorrect  balance.  —  First,  by  substitu- 
tion. Put  the  article  to  be  weighed  in  one  pan  of  the  balance  and  counter- 
poise it  by  any  convenient  heavy  articles  placed  on  the  other  pan. 
Remove  the  article  to  be  weighed  and  substitute  for  it  standard  weights 
until  equipoise  is  again  established.  The  amount  of  these  weights  is  the 
weight  of  the  article. 

Second,  by  transposition.  Determine  the  apparent  weight  of  the 
article  as  usual,  then  its  apparent  weight  after  transposing  the  article  and 
the  weights.  If  the  difference  is  small,  add  half  the  difference  to  the 
smaller  of  the  apparent  weights  to  obtain  the  true  weight.  If  the  differ- 
ence is  2  per  cent  the  error  of  *  his  method  is  1  part  in  10,000.  For  larger 
differences,  or  to  obtain  a  perfectly  accurate  result,  multiply  the  two 
apparent  weights  together  and  extract  the  square  root  of  the  product. 

Circular  Measure. 

60  seconds,  *  =  1  minute,  '. 
60  minutes, '  =  1  degree,  °. 
90  degrees       =  1  quadrant. 
380  =  circumference. 

Arc  of  angle  of  57.3°,  or  360°  •*•  6.2832  =  1  radian  —  the  arc  whose  length 
is  equal  to  the  radius. 

Time. 

60  seconds  =  1  minute. 
60  minutes  =  1  hour. 
24  hours       =  1  day. 

7  days        =  1  week. 
365  days,  5  hours,  48  minutes,  48  seconds  «»  1  year. 

By  the  Gregorian  Calendar  every  year  whose  number  is  divisible  by  4 
is  a  leap  year,  and  contains  366  days,  the  other  years  containing  365  days, 
except  that  the  centesimal  years  are  leap  years  only  when  the  number  of 
the  year  is  divisible  by  400. 

The  comparative  values  of  mean  solar  and  sidereal  time  are  shown  by 
the  following  relations  according  to  Bessel: 

365.24222  mean  solar  days  =  366.24222  sidereal  days,  whence 
1  mean  solar  day  =  1.00273791  sidereal  days; 

1  sidereal  day  =  0.99726957  mean  solar  day; 
24  hours  mean  solar  time  =  24*  3    56«.555  sidereal  time; 
24  hours  sidereal  time  =  23*  56*n  4«.091  mean  solar  time, 

whence  1  mean  solar  day  is  3»  55«.91  longer  than  a  sidereal  day,  reckoned 
in  mean  solar  time. 

BOARD  AND  TIMBER  MEASURE. 

Board  Measure. 

In  board  measure  boards  are  assumed  to  be  one  inch  in  thickness.  To 
obtain  the  number  of  feet  board  measure  (B.  M.)  of  a  board  or  stick  of 
square  timber,  multiply  together  the  length  in  feet,  the  breadth  in  feet, 
and  the  thickness  in  inches. 

To  compute  the  measure  or  surface  in  square  feet.  —  When  all 
dimensions  are  in  feet,  multiply  the  length  by  the  breadth,  and  the  prod- 
uct will  give  the  surface  required. 

When  either  of  the  dimensions  are  in  inches,  multiply  as  above  and 
divide  the  product  by  12. 

When  all  dimensions  are  in  inches,  multiply  as  before  and  divide  product 
by  144. 

Timber  Measure. 

To  compute  the  volume  of  round  timber.  —  When  all  dimensions 
are  in  feet,  multiply  the  length  by  one  quarter  of  the  product  of  the  mean 


WEIGHTS   AND   MEASURES. 


21 


girth  and  diameter,  and  the  product  will  give  the  measurement  in  cubic 
feet.  When  length  is  given  in  feet,  and  girth  and  diameter  in  inches 
divide  the  product  by  144;  when  all  the  dimensions  are  in  inches,  divide 
by  1728. 

To  compute  the  volume  of  square  timber.  —  When  all  dimensions 
are  in  feet,  multiply  together  the  length,  breadth,  and  depth;  the  product 
will  be  the  volume  in  cubic  feet.  When  one  dimension  is  given  in  inches, 
divide  by  12;  when  two  dimensions  are  in  inches,  divide  by  144:  when  all 
three  dimensions  are  in  inches,  divide  by  1728. 

Contents  in  Feet  of  Joists,  Scantling,  and  Timber, 

Length  in  Feet. 


Size. 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

Feet  Board  Measure. 


2X4 

8 

9 

11 

12 

13 

15 

16 

17 

19 

20 

2X6 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

2X8 

16 

19 

21 

24 

27 

29 

32 

35 

37 

40 

2  X  10 

20 

23 

27 

30 

33 

37 

40 

43 

47 

50 

2  X  12 

24 

28 

32 

36 

40 

44 

48 

52 

56 

60 

2  X  14 

28 

33 

37 

42 

47 

51 

56 

61 

65 

70 

3X8 

24 

28 

32 

36 

40 

44 

48 

52 

56 

60 

3  X  10 

30 

35 

40 

45 

50 

55 

60 

65 

70 

75 

3  X  12 

36 

42 

48 

54 

60 

66 

72 

78 

84 

90 

3  X  14 

42 

49 

56 

63 

70 

77 

64 

91 

98 

105 

4X4 

16 

19 

21 

24 

27 

29 

32 

35 

37 

40 

4X6 

24 

28 

32 

36 

40 

44 

43 

52 

56 

60 

4X8 

32 

37 

43 

43 

53 

59 

64 

69 

75 

80 

4  X  10 

40 

47 

53 

60 

67 

73 

80 

87 

93 

100 

4  X  12 

48 

56 

64 

72 

80 

83 

96 

104 

112 

120 

4  X  14 

56 

65 

75 

84 

93 

103 

112 

121 

131 

140 

6X6 

36 

42 

43 

54 

60 

66 

72 

78 

84 

90 

6X8 

48 

56 

64 

72 

80 

83 

96 

104 

112 

120 

6  X  10 

60 

70 

80 

90 

100 

110 

120 

130 

140 

150 

6  X  12 

72 

84 

96 

108 

120 

132 

144 

156 

168 

180 

6X  14 

84 

98 

112 

126 

140 

154 

168 

182 

196 

210 

8X8 

64 

75 

85 

96 

107 

117 

128 

139 

149 

160 

8  X  10 

80 

93 

107 

120 

133 

147 

160 

173 

187 

200 

8  X  12 

96 

112 

128 

144 

160 

176 

192 

208 

224 

240 

8  X  14 

112 

131 

149 

168 

187 

205 

224 

243 

261 

280 

10  X  10 

100 

117 

133 

150 

167 

183 

200 

217 

233 

250 

10  X  12 

120 

140 

160 

180 

200 

220 

240 

260 

2ttO 

300 

10  X  14 

140 

163 

187 

210 

233 

257 

280 

303 

327 

350 

12  X  12 

144 

168 

192 

216 

240 

264 

288 

312 

336 

360 

12  X  14 

168 

196 

224 

252 

280 

308 

336 

364 

392 

420 

14  X  14 

196 

229 

261 

294 

327 

359 

392 

425 

457 

490 

FRENCH  OB  METRIC  MEASURES. 

The  metric  unit  of  length  is  the  metre  =  39.37  inches. 

The  metric  unit  of  weight  is  the  gram  =  15.432  grains. 

1  he  following  prefixes  are  used  for  subdivisions  and  multiples:  Milli  = 
Viooq,  Centi  =  Vioo,  Deci  =  1/10,  Deca  =  10,  Hecto  =  100,  Kilo  =  1000. 
«i.yna  =  10,000. 


22  ARITHMETIC. 

FRENCH  AND  BRITISH  (AND  AMERICAN) 
EQUIVALENT  MEASURES. 

Measures  of  Length. 

FRENCH.  BRITISH  and  U.  S. 

1  metre  =  39.37  inches,  or  3.28083  feet,  or  1.09361  yards. 

0.3048  metre  =  1  foot. 

1  centimetre    =  0.3937  inch. 
2.54  centimetres  =  1  inch. 

1  millimetre    =  0.03937  inch,  or  1  /25  inch,  nearly. 
25.4  millimetres  =  1  inch. 

1  kilometre      =  1093. Gl  yards,  or  0.62137  mile. 

Of  Surface 

FRENCH  BRITISH  and  U.  S. 

1  omiarp  mptrp  -  j  10.7639  square  feet. 

~  1     1.196  square  yards. 

0.836  square  metre  =  1  square  yard. 

0.0929  square  metre  =  1  square  foot. 

1  square  centimetre          =  0. 15500  square  inch. 
6.452  square  centimetres        =  1  square  inch. 

1  square  millimetre  =  0.00155  sq.  in.  =  1973.5  circ.  mils. 

645.2  square  millimetres         =  1  square  inch. 

1  centiare  =  1  sq.  metre  =  10.764  square  feet. 

1  are  =  1  sq.  decametre  =  1076.41 

1  hectare  =  100  ares         =  107641  =  2.4711  acres. 

1  sq.  kilometre  =  0.386109  sq.  miles  =  247.11 

1  sq.  myriametre  =  38.6109 

Of  Volume 

FRENCH.  BRITISH  and  U.  S. 

i  miVnV  rnpfro  J  35.314  cubic  feet, 

1  cubic  metre  =  -j     13QS  cubic  yards 

0.7645  cubic  metre  =  1  cubic  yard. 

0.02832  cubic  metre  =  1  cubic  foot. 

1  oiibio  rlpHmptrP     -  i  61.0234  cubic  inches. 
1     0.035314  cubic  foot. 
28.32  cubic  decimetres    =  1  cubic  foot. 

1  cubic  centimetre    =  0.061  cubic  inch. 
16.387  cubic  centimetres  =  1  cubic  inch. 
1  cubic  centimetre  =  1  millilitre  =    0.061  cubic  inch. 
1  decilitre  =6.102      " 

1  litre  =  1  cubic  decimetre  =  61.0234    '  =  1.05671 

quarts,  U.  S. 

1  hectolitre  or  decistere  =  3.5314  cubic  feet   =  2.8375  bu.,  U.  S. 

1  stere,  kilolitre,  or  cubic  metre  =  1.308  cubic  yards  =  28.37  bu., 

Of  Capacity 

FRENCH.  BRITISH  and  U.  S. 

f  6 1.0234  cubic  inches. 
oiil'gaUoi?  (American), 
2.202  pounds  of  water  at  62°  F. 
28.317  litres  =  1  cubic  foot. 

4.543  litres  =  1  gallon  (British). 

3.785  litres  =  1  gallon  (American). 

Of  Weight. 

FRENCH.  BRITISH  and  U.  S. 

1  gramme  =  15,432  grains. 

0.0648  gramme  =  1  grain. 

1  kilogramme  =  2.204622  pounds. 

0.4536  kilogramme  =  1  pound. 

1  tonne  or  metric  ton  I  =  j  0.9842  ton  of  2240  pounds. 
1000  kilogrammes  f  =  j  22G4. 6  pounds. 

1.016  metric  tons  «    1  ton  of  2240  pounds. 


WEIGHTS  AND  MEASURES. 


23 


Mr.  O.  H.  Titmann,  in  Bulletin  No.  9  of  the  U.  S.  Coast  and  Geodetic 
Survey,  discusses  the  work  of  various  authorities  who  have  compared  the 
yard  and  the  metre,  and  by  referring  all  the  observations  to  a  common 
standard  has  succeeded  in  reconciling  the  discrepancies  within  very 
narrow  limits.  The  following  are  his  results  for  the  number  of  inches  in  a 
metre  according  to  the  comparisons  of  the  authorities  named:  1817. 
Hassler,  39.36994  in.  1818.  Kater,  39.36990  in.  1835.  Baily,  39.36973 
in.  1866.  Clarke,  39.36970  in.  1885.  Comstock,  39.36984  in.  The  mean 
of  these  is  39.36982  in. 

The  value  of  the  metre  is  now  denned  in  the  U.  S.  laws  as  39.37  inches. 

French  and  British  Equivalents  of  Compound  Units. 

FRENCH.  BRITISH. 

gramme  per  square  millimetre  =        1.422  Ibs.  per  sq.  in. 

kilogramme  per  square     '  =  1422.32 

centimetre  =      14.223  " 

.0335  kg.  per  sq.  cm.  =  1  atmosphere  =      14.7       "      "     "     " 

0.070308  kilogramme  per  square  centimetre  =        1  Ib.  per  square  inch. 

kilogrammetre  =        7.2330  foot-pounds. 

gramme  per  litre  =  0.062428  Ib.  per  cu.  ft.  =  58.349  grains  per  U.  S  gal. 

of  water  at  62°  F. 
1  grain  per  U.  S.  gallon=l  part  in  58,349        =  1.7138   parts   per    100,000 

—  0.017138  grammes  per  litre. 

METRIC  CONVERSION  TABLES. 

The  following  tables,  with  the  subjoined  memoranda,  were  published 
in  1890  by  the  United  States  Coast  and  Geodetic  Survey,  office  of  standard 
weights  and  measures,  T.  C.  Mendenhall,  Superintendent. 


- 


Tables  for  Converting  TJ.  S.  Weights  and  Measures  — 
Customary  to  Metric. 

LINEAR. 


Inches  to  Milli- 
metres. 

Feet  to  Metres. 

Yards  to  Metres. 

Miles  to  Kilo- 
metres. 

2  = 

3  = 
4  = 
5  = 

25.4001 
50.8001 
76.2002 
101.6002 
127.0003 

0.304801 
0.609601 
0.914402 
1.219202 
1.524003 

0.914402 
1  .828804 
2.743205 
3.657607' 
4.572009 

1.60935 
3.21869 
4.82804 
6.43739 
8.04674 

8  = 

152.4003 
177.8004 
203.2004 
228.6005 

1.828804 
2.133604 
2.438405 
2:743205 

5.486411 
6.400813 
7.315215 
8.229616 

9.65608 
11.26543 
12.87478 
14.48412 

SQUARE. 


Square  Inches  to 
Square  Centi- 
metres. 

Square  Feet  to 
Square  Deci- 
metres. 

Square  Yards  to 
Square  Metres. 

Acres  to 
Hectares. 

K 

1  = 

6.452 

9.290 

0.836 

04047 

2  = 

12.903 

18.581 

1.672 

0.8094 

3  = 

19.355 

27.871 

2.508 

1.2141 

A    

25.807 

37.161 

3.344 

1.6187 

5  = 

32.258 

46.452 

4.181 

2.0234 

6  = 

38.710 

55.742 

5.017 

2.4281 

7  = 

45.161 

65.032 

5.853 

2.8328 

8  = 

51.613 

74.323 

6.689 

3.2375 

9  = 

58.065 

83.613 

7.525 

3.6422 

ARITHMETIC. 


CUBIC. 


Cubic  Inches  to 
Cubic  Centi- 
metres. 

Cubic  Feet  to 
•Cubic  Metres. 

Cubic  Yards  to 
Cubic  Metres. 

Bushels  to 
Hectolitres. 

Ui-UUJ  Si- 
ll II  II  II  11 

16.387 
32.774 
49.161 
65.549 
81.936 

0.02832 
0.05663 
0.08495 
0.11327 
0.14158 

0.765 
1.529 
2.294 
3.058 
3.823 

0.35242 
0.70485 
1.05727 
1  .40969 
1.76211 

6  = 

8  = 

98.323 
114.710 
131.097 
147.484 

0.16990 
0.19822 
0.22654 
0.25485 

4.587 
5.352 
6.116 
6.881 

2.11454 
2.46696 
2.81938 
3.17181 

CAPACITY. 


Fluid  Dracnms 
to  Millilitres  or 
Cubic  Centi- 
metres. 

Fluid  Ounces  to 
Millilitres  . 

Quarts  to  Litres. 

Gallons  to 
Litres. 

1  = 

2  = 
3  = 
4  = 
5  = 

6  = 

8  = 
9  = 

3.70 
7.39 
11.09 
14.79 
18.48 

22.18 
25.88 
29.57 
33.28 

29.57 
59.15 
88.72 
118.30 
147.87 

177.44 
207.02 
236.59 
266.16 

0.94636 
1  .89272 
2.83908 
3.78544 
4.73180 

5.67816 
6.62452 
7.57088 
8.51724 

3.78544 
7.57088 
11.35632 
15.14176 
18.92720 

22.71264 
26.49808 
30.28352 
34.06896 

WEIGHT. 


Grains  to  Milli- 
grammes. 

Avoirdupois 
Ounces  to 
Grammes. 

Avoirdupois 
Pounds  to  Kilo- 
grammes. 

Troy  Ounces  to 
Grammes. 

1  =» 

2  

4  = 
5  = 

6  = 

8-= 
9- 

64.7989 
129.5978 
194.3968 
259.1957 
323.9946 

388.7935 
453.5924 
518.3914 
583.1903 

28.3495 
56.6991 
85.0486 
113.3981 
141.7476 

170.0972 
198.4467 
226.7962 
255.1457 

0.45359 
0.90719 
1  .36078 
1.81437 
2.26796 

2.72156 
3.17515 
3.62874 
4.08233 

31.10348 
62.20696 
93.31044 
124.41392 
155.51740 

186.62089 
217.72437 
248.82785 
279.93133 

1  chain   =    20.11 69  metres. 
1  square  mile   =   259  hectares. 
1  fathom  =    1 .829  metres. 
1  nautical  mile  =    1853.27  metres. 
1  foot  =   0.304801  metre. 
1  avoir,  pound   =   453.5924277  gram. 
15432.35639  grains    =    1  kilogramme. 


METRIC   CONVERSION   TABLES. 


25 


Tables  for  Converting  U.  S.  Weights  and  Measures — 
Metric  to  Customary. 

LINEAR. 


Metres  to 
Inches. 

Metres  to 
Feet. 

Metres  to 
Yards. 

Kilometres  to 
Miles. 

1  = 

2  = 

4  = 
5  = 

39.3700 
78.7400 
118.1100 
157.4800 
196.8500 

3.28083 
6.56167 
9.84250 
13.12333 
16.40417 

1.093611 
2.187222 
3.280833 
4.374444 
5.468056 

0.62137 
1  .24274 
1.86411 
2.48548 
3.10685 

1: 

236.2200 
275.5900 
314.9600 
354.3300 

19.68500 
22.96583 
26.24667 
29.52750 

6.561667 
7.655278 
8.748889 
9.842500 

3.72822 
4.34959 
4.97096 
5.59233 

SQUARE. 


Square  Centi- 
metres to 
Square  Inches. 

Square  Metres 
to  Square  Feet. 

Square  Metres 
to  Square  Yards. 

Hectares  to 
Acres. 

1  = 

0.1550 

10.764 

1.196 

2.471 

2  = 

0.3100 

21.528 

2.392 

4.942 

3  = 

0.4650 

32.292 

3.588 

7.413 

4  = 

0.6200 

43.055 

4.784 

9.884 

5  = 

0.7750 

53.819 

5.980 

12.355 

6  = 

0.9300 

64.583 

7.176 

14826 

7  = 

1.0850 

75.347 

8.372 

17.297 

8  = 

1.2400 

86.111 

9.563 

19.768 

9  = 

1.3950 

96.874 

10.764 

22.239 

CUBIC. 


Cubic  Centi- 
metres to  Cubic 
Inches. 

Cubic  Deci- 
metres to  Cubic 
Inches. 

Cubic  Metres  to 
Cubic  Feet. 

Cubic  Metres  to 
Cubic  Yards. 

1  = 

0.0610 

61.023 

35.314 

1.308 

2  - 

0.1220 

122.047 

70.629 

2.616 

3  = 

0.1831 

183.070 

105.943 

3.924 

4  = 

0.2441 

244.093 

141.258 

5.232 

5  = 

0.3051 

305.117 

176.572 

6.540 

6  = 

0.3661 

366.140 

211.887 

7.848 

•j  

0.4272 

427.163 

247.201 

9.156 

8  = 

0.4882 

488.187 

282.516 

10.464 

9  = 

0.5492 

549.210 

317.830 

11.771 

CAPACITY. 


Milhlitres  or 
Cubic  Centi- 
metres toFluid 

Centimetres 
to  Fluid 
Ounces. 

Litres  to 
Quarts. 

Dekalitres 
to 
Gallons. 

Hektolitres 
to 
Bushels. 

Drachms. 

1  = 

0.27 

0.338 

1.0567 

2.6417 

2.8375 

2  = 

0.54 

0.676 

2.1134 

5.2834 

5.6750 

3  = 

0.81 

1.014 

3.1700 

7.9251 

8.5125 

4  = 

1.08 

1.352 

4.2267 

10.5668 

11.3500 

5  = 

1.35 

1.691 

5.2834 

13.2085 

14.1875 

6  = 

1.62 

2.029 

6.3401 

15.8502 

17.0250 

j  

1.89 

2.363 

7.3968 

18.4919 

.  19.8625 

8  = 

2.16 

2.706 

8.4534 

21.1336 

22.7000 

9  = 

2.43 

3.043 

9.5101 

23.7753 

25.5375 

26 


ARITHMETIC. 
WEIGHT. 


Milligrammes 
to  Grains. 

Kilogrammes 
to  Grains. 

Hectogrammes 
(  1  00  grammes) 
to  Ounces  Av. 

Kilogrammes 
to  Pounds 
Avoirdupois. 

1  = 

2  = 
3  = 
4  = 
5  = 

0.01543 
0.03086 
0.04630 
0.06173 
0.07716 

15432.36 
30864.71 
46297.07 
61729.43 
77161.78 

3.5274 
7.0548 
10.5822 
14.1096 
17.6370 

2.20462 
4.40924 
6.61386 
8.81849 
11.02311 

6  = 

•j  

8  = 
9  = 

0.09259 
0.10803 
0.12346 
0.13839 

92594.14 
108026.49 
123458.85 
138891.21 

21.1644 
24.6918 
28.2192 
31.7466 

13.22773 
15.43235 
17.63697 
19.84159 

Quintals  to 
Pounds  Av. 

Milliers  or  Tonnes  to 
Pounds  Av. 

Grammes  to  Ounces. 
Troy. 

1  =, 

220.46 

2204.6 

0.03215 

2  = 

440.92 

4409.2 

0.06430 

3  = 

661.38 

6613.8 

0.09645 

4  = 

881.84 

8818.4 

0.12860 

5  - 

1102.30 

11023.0 

0.16075 

6  = 

1322.76 

13227.6 

0.19290 

7  - 

1543.22 

15432.2 

0  22505 

8== 

1763.68 

17636.8 

0.25721 

9  = 

1984.14 

19841.4 

0.28936 

The  British  Avoirdupois  pound  was  derived  from  the  British  standard 
Troy  pound  of  1758  by  direct  comparison,  and  it  contains  7000  grains  Troy. 

The  grain  Troy  is  therefore  the  same  as  the  grain  Avoirdupois,  and  the 
pound  Avoirdupois  in  use  in  the  United  States  is  equal  to  the  British 
pound  Avoirdupois. 

By  the  concurrent  action  of  the  principal  governments  of  the  world  an 
International  Bureau  of  Weights  and  Measures  has  been  established  near 
Paris. 

The  International  Standard  Metre  is  derived  from  the  Metre  des 
Archives,  and  its  length  is  defined  by  the  distance  between  two  lines  at  0° 
Centigrade,  on  a  platinum-iridium  bar  deposited  at  the  International 
Bureau. 

The  International  Standard  Kilogramme  is  a  mass  of  platinum-indium 
deposited  at  the  same  place,  and  its  weight  in  vacua  is  the  same  as  that  of 
the  Kilogramme  des  Archives. 

Copies  of  these  international  standard  weights  and  measures  are 
deposited  in  the  office  of  the  United  States  Bureau  of  Standards. 

The  litre  is  equal  to  a  cubic  decimetre  of  water,  and  it  is  measured  by 
the  quantity  of  distilled  water  which,  at  its  maximum  density,  will 
counterpoise  the  standard  kilogramme  in  a  vacuum;  the  volume  of  such 
a  quantity  of  water  being,  as  nearly  as  has  been  ascertained,  equal  to  a 
cubic  decimetre. 

The  metric  system  was  legalized  in  the  United  States  in  1866.  Many 
attempts  were  made  during  the  50  years  following  to  have  the  U.  S. 
Congress  pass  laws  to  make  the  metric  system  the  legal  standard,  but  they 
have  all  failed.  Similar  attempts  in  Great  Britain  have  also  failed.  For 
arguments  for  and  against  the  metric  system  see  the  report  of  a  committee 
of  the  American  Society  of  Mechanical  Engineers,  1903,  Vol.  24. 


WEIGHTS    AND   MEASURES.  27 

COMPOUND  UNITS. 

Measures  of  Pressure  and  Weight. 

One  pound  force  (or  pressure)  =  the  force  exerted  by  gravity  on  1  Ib. 
of  matter  at  a  place  where  the  acceleration  due  to  gravity  is  32.1740 
feet-per-second  per  second;  that  is  (very  nearly)  the  force  of  gravity  on 
1  Ib.  of  matter  at  latitude  45°  at  the  sea  level. 


1  Ib.  per  square  inch 


144  Ib.  per  square  foot. 

2.0355  in.  of  mercury  at  32°  P. 
2.0416   "     "          "          "62°F. 
2.309  ft.  of  water  at  62°  F. 
27.71  ins.    "        "       "62°  F. 


j     0.1276  in.  of  mercury  at  62°  F. 
1  ounce  per  sq.  in.  1.732  in.  of  water  at  62°  F. 


2116.3  Ib.  per  square  foot. 
I  33.947  ft.  of  water  at  62°  F. 


;i4.71b.  per  sqJn.)  -  j  ™%i>21in,  ofm^curfat  32' F, 


760  millimetres  of  mercury  at  32°  F. 
i    0.03609  Ib.  or  .5774  oz.  per  sq.  in. 
1  inch  of  water  at  62°  F.  =  •<    5.196  Ib.  per  square  foot. 

0.0735  in.  of  mercury  at  62°  F. 


1  foot  of  water  at  62°  F.  -          ™  »   P 


_j 

H 


0.491  Ib.  or  7.86  oz.  per  sq.  in. 

1  inch  of  mercury  at  62°  F.       =  -{    1.134  ft.  of  water  at  62°  F. 

(  13.61  in.  of  water  at  62°  F. 


Weight  of  One  Cubic  Foot  of  Pure  Water. 

At  32°  F.  (freezing-point) 62.418  Ib. 

"   39.1°  F.  (maximum  density) 62.425    ' 

"   62°  F.  (standard  temperature)  in  vacuo 62.355    " 

"    212°  F.  (boiling-point,  under  1  atmosphere) 59.76 

American  gallon  =  231       cubic  ins.  of  water  at  62°  F.   =  8.3356  Ib. 
British  "       =  277.274  "  "       "      "        "        =  10  Ib. 

Weight  of  1  cu.  ft.  of  air-free  distilled  water  at  62°,  weighed  in  air  at 
62°  with  brass  weights  of  8. 4  density  =  62.287  Ib.  =  8.3267  Ib.  per  U.  S. 
gallon. 

Weight  and  Volume  of  Air. 

1  cubic  ft.  of  air  at  32°  F.  and  atmospheric  pressure  weighs  0.080728  Ib. 
i  tt.  «„  v.~'~i~4-    f    •„    +.  ooo  ™         i  0.0005606  Ib.  per  sq.  in. 

.  in  height  of  air  at  3.  \  0.015534  inches  of  water  at  62°  F. 

For  air  at  any  other  temperature  T°  Fahr.  multiply  by  492  -=-  (460  +  T). 
1  Ib.  pressure  per  sq.  ft.  =       12.387  ft.  of  air  at  32°  F. 

1    "  "    sq.  in.  =  1784.          "    "    "     " 

1  inch  of  water  at  62°  F.          =      64.37      "    "    "     " 
For  air  at  any  other  temperature  multiply  by  (460  +  T)  -~  492. 

At  any  fixed  temperature  the  weight  of  a  given  volume  is  proportional 
to  the  absolute  pressure. 

Measures  of  Work,  Power,  and  Duty. 

Unit  of  work. — One  foot-pound,  i.e.,  a  pressure  of  one  pound  exerted 
through  a  space  of  one  foot. 

Horse-power. — The  rate  of  work.  Unit  of  horse-power  =  33,000 
f t.-lb.  per  minute,  or  550  ft.-lb.  per  second  =  1 ,980,000  ft.-lb.  per  hour. 

Heat  unit.  =  heat  required  to  raise  1  Ib.  of  water  1°  F.  (see  page  560). 

00    (")(")() 

Horse-power  expressed  in  heat-units  =       '      ~  =  42.442  heat-units  per 

minute  =  0.7074  heat-unit  per  second  =  2546.5  heat  units  per  hour. 
1  Ib.  of  fuel  per  H.P.  per  hour  =  1,980,000  ft.-lb.  per  Ib.  of  fuel. 
1,000,000  ft.-lb.  per  Ib.  of  fuel  =  1.98  Ib.  of  fuel  per  H.P.  per  hour. 

5280       ^2 
Velocity. — Feet  per  second  =  %QQQ  =  Is  x  miles  Per  hour. 

tons  per  mile  =  2240  =  14  lb<  per  yarci  (sin£le  rail)  • 


28 


ARITHMETIC. 


WIRE  AND  SHEET-METAL  GAUGES  COMPARED. 


•8^ 

<jj  §3 
*§  § 

So® 

Irl 

.po 

S  *  5s 

•gi° 

ing's  and 
iburn  & 
:'s  Gauge. 

» 

—  «  ajo 

-  O  o 
J§  a)-22 

yt'G  nj 
%«% 

!"2  § 
^B°  . 

<gl|i 

Ui 
l«£a- 

;s  o>  c^05 
«  <*§  cS 

9 

|a 

S  c8 

go 

g-2  ^ 
oc  £ 

Sg£ 

|2& 

•§"§  ® 

||| 

J-li 

pill 

3  ™  c8 

*e£i 
p  £s 

1° 

inch. 

inch. 

inch. 

inch. 

inch. 

inch. 

inch. 

0000000 

.49 

.500 

.6666 

.5 

7/o 

000000 

.46 

464 

.625 

.469 

6,0 

00000 

.43 

.432 

.5883 

.438 

5/o 

0000 

.454 

.46 

.393 

.4 

.406 

000 

.425 

.40964 

.362 

.372 

.500 

.375 

3/2 

00 

.38 

.3648 

.331 

.348 

.4452 

344 

2/n 

0 

.34 

.32486 

.307 

.324 

.3964 

.313 

0 

.3 

.2893 

.283 

227 

.3 

.3532 

.281 

1 

2 

.284 

.25763 

.263 

.219 

.276 

.3147 

266 

2 

3 

.259 

.22942 

.244 

.212 

.252 

.2804 

.25 

3 

4 

.238 

J20431 

.225 

.207 

.232 

.250 

.234 

4 

5 

.22 

.18194 

.207 

.204 

.212 

.2225 

,219 

5 

6 

.203 

.16202 

.192 

.201 

.192 

.1981 

.203 

6 

7 

.18 

.14428 

.177 

.199 

.176 

.1764 

.188 

7 

6 

.165 

.12849 

.162 

.197 

.16 

.1570 

.172 

8 

9 

.148 

.11443 

.148 

.194 

.144 

.1398 

.156 

9 

10 

.134 

.10189 

.135 

.191 

.128 

.1250 

.141 

10 

11 

.12 

.09074 

.12 

.188 

.116 

.1113 

.125 

11 

12 

.109 

.08081 

.105 

.185 

.104 

.0991 

.109 

12 

13 

.095 

.07196 

.092 

.182 

.092 

.0882 

.094 

13 

14 

.033 

.06403 

,08 

.180 

.08 

.0785 

078 

14 

15 

.072 

.05707 

.072 

.178 

.072 

.0699 

.07 

15 

16 

.065 

.05082 

.063 

.175 

.064 

.0625 

.0625 

16 

17 

.058 

.04526 

.054 

.172 

.056 

.0556 

.0563 

17 

18 

.049 

.0403 

.047 

.168 

.048 

.0495 

.05 

18 

19 

.042 

.03589 

.041 

164 

.04 

.0440 

.0433 

19 

20 

.035 

.03196 

.035 

.161 

.036 

.0392 

.0375 

20 

21 

.032 

.02846 

.032 

.157 

.032 

.0349 

.0344 

21 

22 

.028 

.02535 

.028 

.155 

.028 

.03125 

.0313 

22 

23 

.025 

.02257 

.025 

.153 

.024 

.02782 

0281 

23 

24 

.022 

.0201 

.023 

.151 

.022 

.02476 

.025 

24 

25 

.02 

.0179 

.02 

.148 

.02 

.02204 

.0219 

25 

26 

.018 

.01594 

.018 

.146 

.018 

.01961 

.0188 

26 

27 

.016 

.01419 

.017 

.143  •  .0164 

.01745 

.0172 

27 

28 

.014 

.01264 

.016 

.139  .0148 

.015625 

.0156 

28 

29 

.013 

.01126 

.015 

.134  .0136 

.0139 

.0141 

29 

30 

.012 

.01002 

.014 

.127  .0124 

.0123 

.0125 

30 

31 

.01 

.00893 

.013 

.120 

.0116 

.0110 

.0109 

31 

32 

.009 

.00795 

.013 

.115 

.0108 

.0098 

.0101 

32 

33 

.008 

.00708 

.011 

.112 

.01 

.0037 

.0094 

33 

34 

.007 

,0063 

.01 

.110 

.0092 

.0077 

.0086 

34 

35 

.005 

.00561 

.0095 

.103 

.0084 

.0069 

.0078 

35 

36 

.004 

.005 

.009 

.106 

.0076 

.0061 

.007 

36 

37 

.00445 

.0085 

.103 

.0068 

.0054 

.0066 

37 

38 

.00396 

.008 

.101 

.006 

.0048 

.0063 

38 

39 

.00353 

.0075 

.099 

.0052 

.0043 

39 

40 

.00314 

.007 

.097 

.0048 

.00386 

40 

41 

.095 

.0044 

.00343 

41 

42 

.092 

.004 

.00306 

42 

43 

.088 

.0036 

.00272 

43 

44 

.085 

.0032 

.00242 

44 

45 

.081 

.0028 

.00215 

45 

46 

.079 

.0024 

.00192 

46 

47 

.077 

.002 

.00170 

47 

•   48 

.075 

.0016 

.00152 

48 

49 

.072 

.0012 

.00135 

49 

50 

.065 

.001 

.00120 

50 

WIRE   AND   SHEET  METAL   GAUGES ,  29 


THE  EDISON  OB  CIRCULAR  MIL  WIRE  GAUGE. 

(For  table  of  copper  wires  by  this  gauge,  giving  weights,  electrical 
resistances,  etc.,  see  Copper  Wire.) 

Mr.  C.  J.  Field  (Stevens  Indicator,  July,  1887)  thus  describes  the  origin 
of  the  Edison  gauge: 

The  Edison  company  experienced  inconvenience  and  loss  by  not  having 
a  wide  enough  range  nor  sufficient  number  of  sizes  in  the  existing  gauges. 
This  was  felt  more  particularly  in  the  central-station  work  in  making 
electrical  determinations  for  the  street  system.  They  were  compelled  to 
make  use  of  two  of  the  existing  gauges  at  least,  thereby  introducing  a 
complication  that  was  liable  to  lead  to  mistakes  by  the  contractors  and 
linemen. 

In  the  incandescent  system  an  even  distribution  throughout  the  entire 
system  and  a  uniform  pressure  at  the  point  of  delivery  are  obtained  by 
calculating  for  a  given  maximum  percentage  of  loss  from  the  potential  as 
delivered  from  the  dynamo.  In  carrying  this  out,  on  account  of  lack  of 
regular  sizes,  it  was  often  necessary  to  use  larger  sizes  than  the  occasion 
demanded,  and  even  to  assume  new  sizes  for  large  underground  conductors. 
The  engineering  department  of  the  Edison  company,  knowing  the  require- 
ments, have  designed  a  gauge  that  has  the  widest  range  obtainable  and 
a  large  number  of  sizes  which  increase  in  a  regular  and  uniform  manner. 
The  basis  of  the  graduation  is  the  sectional  area,  and  the  number  of  the 
wire  corresponds.  A  wire  of  100,000  circular  mils  area  is  No.  100;  a  wire 
of  one  half  the  size  will  be  No.  50;  twice  the  size  No.  200. 

In  the  older  gauges,  as  the  number  increased  the  size  decreased.  With 
this  gauge,  however,  the  number  increases  with  the  wire,  and  the  number 
multiplied  by  1000  will  give  the  circular  mils. 

The  weight  per  mil-foot,  0.00000302705  pounds,  agrees  with  a  specific 
gravity  of  8.889,  which  is  the  latest  figure  given  for  copper.  The  ampere 
capacity  which  is  given  was  deduced  from  experiments  made  in  the  com- 
pany's laboratory,  and  is  based  on  a  rise  of  temperature  of  50°  F.  in  the 
wire. 

In  1893  Mr.  Field  writes,  concerning  gauges  in  use  by  electrical  engineers: 

The  B.  and  S.  gauge  seems  to  be  in  general  use  for  the  smaller  sizes,  up 
to  100,000  c.m.,  and  in  some  cases  a  little  larger.  From  between  one  and 
two  hundred  thousand  circular  mils  upwards,  the  Edison  gauge  or  its 
equivalent  is  practically  in  use,  and  there  is  a  general  tendency  to  desig- 
nate all  sizes  above  this  in  circular  mils,  specifying  a  wire  as  200,000, 
400,000,  500,000,  or  1,000,000  C.M. 

In  the  electrical  business  there  is  a  large  use  of  copper  wire  and  rod  and 
other  materials  of  these  large  sizes,  and  in  ordering  them,  speaking  of 
them,  specifying,  and  in  every  other  use,  the  general  method  is  to  simply 
specify  the  circular  milage.  I  think  it  is  going  to  be  the  only  system  in 
the  future  for  the  designation  of  wires,  and  the  attaining  of  it  means 
practically  the  adoption  of  the  Edison  gauge  or  the  method  and  basis  of 
this  gauge  as  the  correct  one  for  wire  sizes. 

THE  U.  S.  STANDARD  GAUGE  FOR  SHEET  AND 
PLATE  IRON  AND  STEEL,   1893. 

There  is  in  this  country  no  uniform  or  standard  gauge,  and  the  same 
numbers  in  different  gauges  represent  different  thicknesses  of  sheets  or 
plates.  This  has  given  rise  to  much  misunderstanding  and  friction 
between  employers  and  workmen  and  mistakes  and  fraud  between  dealers 
and  consumers. 

An  Act  of  Congress  in  1893  established  the  Standard  Gauge  for  sheet 
Iron  and  steel  which  is  given  on  the  next  page.  It  is  based  on  the  fact  that 
a  cubic  foot  of  iron  weighs  480  pounds. 

A  sheet  of  iron  1  foot  square  and  1  inch  thick  weighs  40  pounds,  or  640 
ounces,  and  1  ounce  in  weight  should  be  1/640  inch  thick.  The  scale  has 
been  arranged  so  that  each  descriptive  number  represents  a  certain 
number  of  ounces  in  weight  and  an  equal  number  of  640ths  of  an  inch  in 
thickness. 

The  law  enacts  that  on  and  after  July  1,  1893,  the  new  gauge  shall  be 
used  in  determining  duties  and  taxes  levied  on  sheet  and  plate  iron  and 
(Continued  on  page  32.} 


30 


ARITHMETIC. 


Edison,  or  Circular  Mil  Gauge  for  Electrical  Wires. 


Gauge 
Num- 
ber. 

Circular 
Mils. 

Diam- 
eter in 
Mils. 

Gauge 
Num- 
ber. 

Circular 
Mils. 

Diam- 
eter in 

Mils. 

Gauge 
Num- 
ber. 

Circular 
Mils. 

Diam- 
eter in 
Mils. 

3 

3,000 

54.78 

70 

70,000 

264.58 

190 

190,000 

435.89 

5 

5,000 

70.72 

75 

75,000 

273.87 

200 

200,000 

447.22 

8 

8,000 

89.45 

80 

80,000 

282.85 

220 

220,000 

469.05 

12 

12,000 

109.55 

85 

85,000 

291.55 

240 

240,000 

489.90 

15 

15,000 

122.48 

90 

90,000 

300.00 

260 

260,000 

509.91 

20 

20,000 

141.43 

95 

95,000 

308.23 

280 

280,000 

529.16 

25 

25,000 

158.12 

100 

100,000 

316.23 

300 

300,000 

547.73 

30 

30,000 

173.21 

110 

110,000 

331.67 

320 

320,000 

565.69 

35 

35,000 

187.09 

120 

120,000 

346.42 

340 

340,000 

583.10 

40 

40,000 

200.00 

130 

130,000 

360.56 

360 

360,000 

600.00 

45 

45,000 

212.14 

140 

140,000 

374.17 

50 

50,000 

223.61 

150 

150,000 

387.30 

55 

55,000 

234.53 

160 

160,000 

400.00 

60 

60,000 

244.95 

170 

170,000 

412.32 

65 

65,000 

254.96 

180 

180,000 

424.27 

Twist  Drill  and  Steel  Wire  Gauge. 

(Manufacturers  Standard) 


No. 

Size. 

No. 

Size. 

No. 

Size. 

No. 

Size. 

No. 

Size. 

No. 

Size. 

inch. 

inch. 

inch. 

inch. 

inch. 

inch. 

1 

0.2280 

14 

0.1820 

27 

0.1440 

40 

0.0980 

53 

0.0595 

67 

0.0320 

2 

.2210 

15 

.1800 

28 

.1405 

41 

.0960 

54 

.0550 

68 

.0310 

.2130 

16 

.1770 

29 

.1360 

42 

.0935 

55 

.0520 

69 

.0292 

4 

.2090 

17 

.1730 

30 

.1285 

43 

.0890 

56 

.0465 

70 

.0280 

5 

.2055 

18 

.1695 

31 

.1200 

44 

.0860 

57 

.0430 

71 

.0260 

6 

.2040 

19 

.1660 

32 

.1160 

45 

.0820 

58 

.0420 

72 

.0250 

7 

.2010 

20 

.1610 

33 

.1130 

46 

.0810 

59 

.0410 

73 

.0240 

8 

.1990 

21 

.1590 

34 

.1110 

47 

.0785 

60 

.0400 

74 

.0225 

9 

.1960 

22 

.1570 

35 

.1100 

48 

.0760 

61 

.0390 

75 

.0210 

10 

.1935 

23 

.1540 

36 

.1065 

49 

.0730 

62 

.0380 

76 

.0200 

11 

.1910 

24 

.1520 

37 

.1040 

50 

.0700 

63 

.0370 

77 

.0180 

12 

.1890 

25 

.1495 

38 

.1015 

51 

.0670 

64 

.0360 

78 

.0160 

13 

.1850 

26 

.1470 

39 

.0995 

52 

.0635 

65 

.0350 

79 

.0145 

66 

.0330 

80 

.0135 

Stubs'  Steel  Wire  Gauge. 

(For  Nos.  1  to  50  see  table  on  page  31.) 


No. 

Size. 

No. 

Size. 

No. 

Size. 

No. 

Size. 

No. 

Size. 

No. 

Size. 

z 

inch. 
.413 

P 

inch. 
.323 

F 

inch. 
.257 

51 

inch. 
.066 

61 

inch. 
.038 

71 

inch. 
.026 

Y 

.404 

O 

.316 

Fi 

.250 

52 

.063 

62 

.037 

72 

.024 

X 

.397 

N 

.302 

D 

.246 

53 

.058 

63 

.036 

73 

.023 

w 

.386 

M 

.295 

0 

.242 

54 

.055 

64 

.035 

74 

.022 

V 

.377 

T, 

.290 

B 

.238 

55 

.050 

65 

.033 

75 

.020 

TT 

.368 

K 

.281 

A 

.234 

56 

.045 

66 

.032 

76 

.018 

T 

.358 

,T 

.277 

1 

(See 

57 

.042 

67 

.031 

77 

.016 

8 

.348 

T 

.272 

to 

{page 

58 

.041 

68 

.030 

78 

.015 

fi 

.339 

H 

.266 

50 

(29 

59 

.040 

69 

.029 

79 

.014 

Q 

.332 

G 

.261 

60 

.039 

70 

.027 

80 

.013 

The  Stubs'  Steel  Wire  Gauge  is  used  in  measuring  drawn  steel  wire  or 
drill  rods  of  Stubs'  make,  and  is  also  used  by  many  makers  of  American 
drill  rods. 


WIRE   AND   SHEET   METAL   GAUGES. 


31 


U.  S.  STANDARD  GAUGE  FOR  SHEET  AND  PLATE 
IRON  AND   STEEL,  1893. 


Number  of 
Gauge. 

Approximate 
Thickness  in 
Fractions  of 
an  Inch. 

**    9 
8  «-a  *  . 

HIP 

fr* 

Approximate 
Thickness 
in 

Millimeters. 

Weight  per 
Square  Foot 
in  Ounces 
Avoirdupois. 

Weight  per 
Square  Foot 
in  Pounds 
Avoirdupois. 

fit 

^§5 

|§S 

^£.2 

Weight  per 
Square  Meter 
in  Kilograms. 

1  Weight  per  Sq.  1 
M  eter  in  Founds! 
Avoirdupois.  | 

0000000 

1-2 

0.5 

12.7 

320 

20. 

9.072 

97.65 

215.28 

000000 

15-32 

0.46875 

1  1  .90625 

300 

18.75 

8.505 

91.55 

201.82 

00000 

7-16 

0.4375 

11.1125 

280 

17.50 

7.938 

85.44 

188.37 

0000 

13-32 

0.40625 

10.31875 

260 

16.25 

7.371 

79.33 

174.91 

000 

3-8 

0.375 

9.525 

240 

15. 

6.804 

73.24 

161.46 

00 

11-32 

0.34375 

8.73125 

220 

13.75 

6.237 

67.13 

148.00 

0 

5-16 

0.3125 

7.9375 

200 

12.50 

5.67 

61.03 

134.55 

1 

9-32 

0.28125 

7.14375 

180 

11.25 

5.103 

54.93 

121.09 

2 

17-64 

0.265625 

6.746875 

170 

10.625 

4.819 

51.88 

114.37 

3 

1-4 

0.25 

6.35 

160 

10. 

4.536 

48.82 

107,64 

4 

15-64 

0.234375 

5.953125 

150 

9.375 

4.252 

45.77 

100.91 

5 

7-32 

0.21875 

5.55625 

140 

8.75 

3.969 

42.72 

94.18 

6 

13-64 

0.203125 

5.159375 

130 

8.125 

3.685 

39.67 

87.45 

7 

3-16 

0.1875 

4.7625 

120 

7.5 

3.402 

36.62 

80.72 

.       8 

11-64 

0.171875 

4.365625 

110 

6.875 

3.118 

33.57 

74.00 

9 

5-32 

0.15625 

3.96875 

100 

6.25 

2.835 

30.52 

67.27 

10 

9-64 

0.140625 

3.571875 

90 

5.625 

2.552 

27.46 

60.55 

11 

1-8 

0.125 

3.175 

80 

5. 

2.268 

24.41 

53.82 

12 

7-64 

0.109375 

2.778125 

70 

4.375 

.984 

21.36 

47.09 

13 

3-32 

0.09375 

2.38125 

60 

3.75 

.701 

18.31 

40.36 

14 

5-64 

0.078125 

1  .984375 

50 

3.125 

.417 

15.26 

33.64 

15 

9-128 

0.0/03125 

1  .7859375 

45 

2.8125 

.276 

13.73 

30.27 

16 

1-16 

0.0625 

1.5875 

40 

2.5 

.134 

12.21 

26.91 

17 

9-160 

0.05625 

1  .42875 

36 

2.25 

.021 

10.99 

24.22 

18 

1-20 

0.05 

1.27 

32 

2. 

0.9072 

9.765 

21.53 

19 

7-160 

0.04375 

1.11125 

28 

.75 

0.7938 

8.544 

18.84 

20 

3-80 

0.0375 

0.9525 

24 

.50 

0.6804 

7.324 

16.15 

21 

1  1-320 

0.034375 

0.873125 

22 

.375 

0.6237 

6.713 

14.80 

22 

1-32 

0.03125 

0.793750 

20 

.25 

0.567 

6.103 

13.46 

23 

9-320 

0.028125 

0.714375 

18 

.125 

0.5103 

5.49 

12.11 

24 

1-40 

0.025 

0.635 

16 

1. 

0.4536 

4.882 

10.76 

25 

7-320 

0.021875 

0.555625 

14 

0.875 

0.3969 

4.272 

9.42 

26 

3-160 

0.01875 

0.47625 

12 

0.75 

0.3402 

3.662 

8.07 

27 

1  1-640 

0.0171875 

0.4365625 

11 

0.6875 

0.3119 

3.357 

7.40 

28 

1-64 

0.015625 

0.396875 

10 

0.625 

0.2835 

3.052 

6.73 

29 

9-640 

0.0140625 

0.3571875 

9 

0.5625 

0.2551 

7746 

6.05 

30 

1-80 

0.0125 

0.3175 

8 

0.5 

0.2268 

2.441 

5.38 

31 

7-640 

0.0109375 

0.2778125 

7 

0.4375 

0.1984 

2.136 

4.71 

32 

13-1280 

0.01015625 

0.25796875 

'  <     61/2 

0.40625 

0.1843 

1.983 

4.37 

(33 

3-320 

0.009375 

0.238125 

6 

0.375 

0.1701 

1.831 

4.04 

34 

11-1280 

0.00859375 

0.21828125 

51/2 

0.34375 

0.1559 

1.678 

3.70 

35 

5-640 

0.0078125 

0.1984375 

5 

0.3125 

0.1417 

1.526 

3.36 

36 

9-1280 

0.00703125 

0.17859375 

4V2 

0.28125 

0.1276 

1.373 

3.03 

37 

17-2560 

0.00664062 

0.16867187 

41/4 

0.26562 

0.1205 

1.297 

2.87 

38 

1-160 

0.00625 

0.15875 

0.25 

0.1134 

1.221 

2.69 

- 


32 


THE   DECIMAL   GAUGE. 


(continued  from  page  29)  steel;  and  that  in  its  application  a  variation  of 
2 1/2  per  cent  either  way  may  be  allowed. 

The  Decimal  Gauge.  —  The  legalization  of  the  standard  sheet- 
metal  gauge  of  1893  and  its  adoption  by  some  manufacturers  of 
sheet  iron  have  only  added  to  the  existing  confusion  of  gauges.  A  joint 
committee  of  the  American  Society  of  Mechanical  Engineers  and  the 
American  Railway  Master  Mechanics'  Association  in  1895  agreed  to 
recommend  the  use  of  the  decimal  gauge,  that  is,  a  gauge  whose  number 
for  each  thickness  is  the  number  of  thousandths  of  an  inch  in  that  thick- 
ness, and  also  to  recommend  "  the  abandonment  and  disuse  of  the  various 
other  gauges  now  in  use,  as  tending  to  confusion  and  error."  A  notched 
gauge  of  oval  form,  shown  in  the  cut  below,  has  come  into  use  as  a  standard 
form  of  the  decimal  gauge. 

In  1904  The  Westinghouse  Electric  &  Mfg.  Co.  abandoned  the  use  of 
gauge  numbers  in  referring  to  wire,  sheet  metal,  etc. 

Weight  of  Sheet  Iron  and  Steel.    Thickness  by  Decimal  Gauge. 


§ 

i 

Weight  per 
Square  Foot 

6 

§ 

m 

Weight  per 
Square  Foot 

3 

.2  • 

J* 

in  Pounds. 

!> 

'§  ' 

"£ 

in  Pounds. 

• 

"OBO 

• 

09 

t>o 

1 

• 

0 

£  d 

JO  ^ 

o. 

O 

E  £ 

-£  -tJ 

"O 

•a 

fehfl 

i 

o^. 

®  ft-^ 

'd 

^  £ 

iS 

o  ~ 

&   4)     • 

00   ft,*^ 

.3 

X  * 

M 

*O 

"i-    ^» 

.5 

o 

X 

$5 

"*"__  r* 

Q 

I 

ft 

0  ft 
M 

135 

02 

Q 

P 

2 
1 

'a)^  3 

£ 

0.002 

1/500 

0.05 

0.08 

0.082 

0.060 

1/16- 

1.52 

2.40 

2.448 

0.004 

1/250 

0.10 

0.16 

0.163 

0.065 

13/200 

1.65 

2.60 

2.652 

0.006 

3/500 

0.15 

0.24 

0.245 

0.070 

7/100 

1.78 

2.80 

2.856 

0.008 

Vl25 

0.20 

0.32 

0.326 

0.075 

3/40 

1.90 

3.00 

3.060 

0.010 

1/100 

0.25 

0.40 

0.408 

0.080 

2/25 

2.03 

3.20 

3.264 

0.012 

3/250 

0.30 

0.48 

0.490 

0.085 

17/200 

2.16 

3.40 

3.468 

0.014 

7/500 

0.36 

0.56 

0.571 

0.090 

9/100 

2.28 

3.60 

3.672 

0.016 

1/64  + 

0.41 

0.64 

0.653 

0.095 

19/200 

2.41 

3.80 

3.876 

0.018 

9/500 

0.46 

0.72 

0.734 

0.100 

1/10 

2.54 

4.00 

4.080 

0.020 

1/50 

0.51 

0.80 

0.816 

0.110 

H/100 

2.79 

4.40 

4.488 

0.022 

H/500 

0.56 

0.88 

0.898 

0.125 

1/8 

3.18 

5.00 

5.100 

0.025 

1/40 

0.64 

.00 

.020 

0.135 

27/200 

3.43 

5.40 

5.508 

0.028 

7/250 

0.71 

.12 

.142 

0.150 

3/20 

3.81 

6.00 

6.120 

0.032 

1/32  + 

0.81 

.28 

.306 

0.165 

33/200 

4.19 

6.60 

6.732 

0.036 

9/250 

0.91 

.44 

.469 

0.180 

9/50 

4.57 

7.20 

7.344 

0.040 

1/25 

1.02 

.60 

.632 

0.200 

1/5 

5.08 

8.00 

8.160 

0.045 

9/200 

1.14 

.80 

.836 

0.220 

n/50 

5.59 

8.80 

8.976 

0.050 

1/20 

1.27 

2.00 

2.040 

0.240 

6.10 

9.60 

9.792 

0.055 

11/200 

1.40 

2.20 

2.244 

0.250 

i/f 

6.35 

10.00 

10.20C 

ALGEBRA.  33 


ALGEBRA. 

Addition.  —  Add  a,  b,  and  —  c.     Ans.  a  4  b  —  c. 
Add   2a  and    -  3a.     Ans.    -  a.     Add   2ab,  -  Sab,  -  c,    -  3c.     Ans, 
-  ab  -  4c.     Add  a2  and  2a.     Ans.  a2  -f  2a. 
Subtraction.  —  Subtract  a  from  b.     Ans.  &  —  a.     Subtract  —  a  from 

—  6.     Ans.   —  b  +  a. 

Subtract  b  +  c  from  a.  Ans.  a  —  6  —  c.  Subtract  3a26  —  9c  from 
4a26  -f  c.  Ans.  a26  +  lOc.  RULE:  Change  the  signs  of  the  subtrahend 
and  proceed  as  in  addition. 

Multiplication.  —  Multiply  a  by  b.  Ans.  ab.  Multiply  ab  by  a  +  b. 
Ans.  a26  +  ab2. 

Multiply  a  4-  b  by  a  4  6.    Ans.  (a  4-6)  (a46)=a24-2a&+62. 

Multiply  —  a  by  —  b.  Ans.  ab.  Multiply  —a  by  b.  Ans.  —  a&. 
Like  signs  give  plus,  unlike  signs  minus. 

Powers  of  numbers.  —  The  product  of  two  or  more  powers  of  any 
number  is  the  number  with  an  exponent  equal  to  the  sum  of  the  powers: 
a2  x  a3  =  a5;  a262  X  ab  =  a363;  -  7ab  X  2ac  =  -  14ft26c. 

To  multiply  a  polynomial  by  a  monomial,  multiply  each  term  of  the 
polynomial  by  the  monomial  and  add  the  partial  products:  (6a  —  36) 
X  3c  =  I8ac  -  96c. 

To  multiply  two  polynomials,  multiply  each  term  of  one  factor  by  each 
term  of  the  other  and  add  the  partial  products:  (5a  —  66)  X  (3a  —  46) 
=  15a2  -  38a6  4-  2462. 

The  square  of  the  sum  of  two  numbers  =  sum  of  their  squares  +  twice 
their  product. 

The  square  of  the  difference  of  two  numbers  =  the  sum  of  their  squares 

—  twice  their  product. 

The  product  of  the  sum  and  difference  of  two  numbers  =  the  difference 
of  their  squares: 

(a  4-  6)2  =  a2  4-  2a6  4-  62;  (a  -  6)2  =  a2  -  2a6  4-  &2; 
(a  4-  6)  X  (a  -  6)  =  a2  -  62. 

The  square  of  half  the  sums  of  two  quantities  is  equal  to  their  product 
plus  the  square  of  half  their  difference:  (^^Y  =  ab  +  (^-bY- 

The  square  of  the  sum  of  two  quantities  is  equal  to  four  times  their 
products,  plus  the  square  of  their  difference:  (a  +  6)2  =  4a6  4-  (a  —  6)2. 

The  sum  of  the  squares  of  two  quantities  equals  twice  their  product, 
plus  the  square  of  their  difference:  a2  +  62  =  2a6  4-  (a  —  6)2. 

The  square  of  a  trinomial  ==  square  of  each  term  4  twice  the  product 
of  each  term  by  each  of  the  terms  that  follow  it:  (a  +  6  4  c)2  =  a2  4  62 
4  c2  4-  2a6  4-  2ac  +  2bc;  (a  —  b  -  c)?  =  -a2  +  62  +  c*  -  2a6-  2ac  +  2bc. 

The  square  of  (any  number  4-  1/2)  =  square  of  the  number  +  the  number 
+  1/4;  =  the  number  X  (the  number  4-  1)  4- 1/4:  (a+  i/2)2  =  a2  4-  a  4-  1/4, 
=  a  (a  +  1)  4- 1/4-  (4l/2)2  =  42  +  4  4- 1/4=4  X  5  +  1/4  =  201/4. 

The  product  of  any  number  4-  1/2  by  any  other  number  +  1/2  =  product 
of  the  numbers  4-  half  their  sum  4- 1/4.  (a  +  i/2)  X  (6  4-  1/2)  =  ab  +  1/2(0  46) 
4  1/4.  4l/2  X  6V2  =  4X64-  i/2(4  4-  6)  4-  V4  =  24  4-  5  4-  1/4  =  29V4. 

Square,  cube,  4th  power,  etc.,  of  a  binomial  a  4-  6. 

(a  4  6)2  =  a2  4-  2a6  4-  62;  (a  4-  6)3  =  a3  4-  3a26  +  3a62  4-  6« 
(a  4-  6)4  =  a4  4-  4a36  4-  6a262  4-  4a63  4-  64. 

In  each  case  the  number  of  terms  is  one  greater  than  the  exponent  of 
the  power  to  which  the  binomial  is  raised. 

2.  In  the  first  term  the  exponent  of  a  is  the  same  as  the  exponent  of  the 
power  to  which  the  binomial  is  raised,  and  it  decreases  by  1  in  each  suc- 
ceeding term. 

3.  6  appears  in  the  second  term  with  the  exponent  1,  and  its  exponent 
increases  by  1  in  each  succeeding  term. 

4.  The  coefficient  of  the  first  term  is  1. 

5.  The  coefficient  of  the  second  term  is  the  exponent  of  the  power  to 
which  the  binomial  is  raised. 


34  ALGEBRA. 


6.  The  coefficient  of  each  succeeding  term  is  found  from  the  next  pre- 
ceding term  by  multiplying  its  coefficient  by  the  exponent  of  a,  and 
dividing  the  product  by  a  number  greater  by  1  than  the  exponent  of  b. 
(See  Binomial  Theorem,  below.) 

Parentheses.  —  When  a  parenthesis  is  preceded  by  a  plus  sign  it  may 
be  removed  without  changing  the  yalue  of  the  expression:  a  +  b  +  (a  + 
b)  =  2a  +  2b.  When  a  parenthesis  is  preceded  by  a  minus  sign  it  may 
be  removed  if  we  change  the  signs  of  all  the  terms  within  the  parenthesis: 
1  —  (a  —  b  —  c)  =  1  —  a  +  b  +  c.  When  a  parenthesis  is  within  a 
parenthesis  remove  the  inner  one  first:  a  —  [6  —  {c  —  (d  —  e)}]  =  a  —  [ft  — 
{c  —  d  +  ej]=  a  -  [b  -  c  +  d  -  e]  =  a  -  b  +  c  —  d  +  e. 

A  multiplication  sign,  X,  has  the  effect  of  a  parenthesis,  in  that  the 
operation  indicated  by  it  must  be  performed  before  the  operations  of 
addition  or  subtraction,  a  4-  b  X  a  +  b  =  a  +  ab  +  b;  while  (a  -f-  6) 
X  (a  +  6)  =  a2  +  2ab  +  62,  and  (a  +  b)  X  a  +  b  =  a2  +  ab  +  b. 

The  absence  of  any  sign  between  two  parentheses,  or  between  a  quan- 
tity and  a  parenthesis,  indicates  that  the  parenthesis  is  to  be  multiplied  by 
the  quantity  or  parenthesis:  a(a  +  b  +  c)  =  a2  +  ab  +  ac. 

Division.  — The  quotient  is  positive  when  the  dividend  and  divisor 
have  like  signs,  and  negative  when  they  have  unlike  signs:  abc  -*-  b  =  ac; 
abc  •*•  —  6  =  —  ac. 

To  divide  a  monomial  by  a  monomial,  write  the  dividend  over  the 
divisor  with  a  line  between  them.  If  the  expressions  have  common  factors, 
remove  the  common  factors: 

azbx       ax     a4  a3       1 

azbx  -*•  aby  =  -r —  =  —  ;  —,  =  a;  -7  =  -^  =  a~~2. 
aby         y      a3         'a6      a2 

To  divide  a  polynomial  by  a  monomial,  divide  each  term  of  the  poly- 
nomial by  the  monomial:  (Sab  —  12ac)  -5-  4a  =  26  —  3c. 

To  divide  a  polynomial  by  a  polynomial,  arrange  both  dividend  and 
divisor  in  the  order  of  the  ascending  or  descending  powers  of  some  common 
letter,  and  keep  this  arrangement  throughout  the  operation. 

Divide  the  first  term  of  the  dividend  by  the  first  term  of  the  divisor,  and 
write  the  result  as  the  first  term  of  the  quotient. 

Multiply  all  the  terms  of  the  divisor  by  the  first  term  of  the  quotient 
and  subtract  the  product  from  the  dividend.  If  there  be  a  remainder, 
consider  it  as  a  new  dividend  and  proceed  as  before:  (a2  —  b2)  -*•  (a  +  b). 

a2  -  62    I  a  +  b. 
a2  +  ab  I  a  —  b. 

-  ab  -~~&~ 

-  ab  -  ft2. 


The  difference  of  two  equal  odd  powers  of  any  two  numbers  is  divisible 
by  their  difference  but  not  by  their  sum: 


The  difference  of  two  equal  even  powers  of  two  numbers  is  divisible  by 
their  difference  and  also  by  their  sum:  (a2  —  b2)  -5-  (a  —  6)  =  a  -4-  6. 

The  sum  of  two  equal  even  powers  of  two  numbers  is  not  divisible  by 
either  the  difference  or  the  sum  of  the  numbers;  but  when  the  exponent 
of  each  of  the  two  equal  powers  is  composed  of  an  odd  and  an  even  factor, 
the  sum  of  the  given  power  is  divisible  by  the  sum  of  the  powers  expressed 
by  the  even  factor.  Thus  x6  +  y6  is  not  divisible  by  x  -f  y  or  by  x  —  y, 
but  is  divisible  by  x2  +  if. 

Simple  equations.  —  An  equation  is  a  statement  of  equality  between 
two  expressions;  as,  a  +  b  =  c  +  d. 

A  simple  equation,  or  equation  of  the  first  degree,  is  one  which  contains 
only  the  first  power  of  the  unknown  quantity.  If  equal  changes  be  made 
(by  addition,  subtraction,  multiplication,  or  division)  in  both  sides  of  an 
equation,  the  results  will  be  equal. 

Any  term  may  be  changed  from  one  side  of  an  equation  to  another, 
provided  its  sign  be  changed:  a+b  =  c+d\a^c+d  —  b.  To  solve 


ALGEBRA.  35 

an  equation  having  one  unknown  quantity,  transpose  all  the  terms  involv- 
ing the  unknown  quantity  to  one  side  of  the  equation,  and- all  the  other 
terms  to  the  other  side;  combine  like  terms,  and  divide  both  sides  by  the 
coefficient  of  the  unknown  quantity. 

Solve  Sx  -  29  -  26  -  3x.     Sx  +  3x  =  29  4-  26;  llx  =  55;  x  =  5,  ans. 

Simple  algebraic  problems  containing  one  unknown  quantity  are  solved 
by  making  x  =  the  unknown  quantity,  and  stating  the  conditions  of  the 
problem  in  the  form  of  an  algebraic  equation,  and  then  solving  the  equa- 
tion. What  two  numbers  are  those  whose  sum  is  48  and  difference  14? 
Let  x  =  the  smaller  number,  re  4-  14  the  greater,  x  +  x  +  14  =  48. 
2x  =  34,  x  =  17;  x  -I-  14  =  31,  ans. 

Find  a  number  whose  treble  exceeds  50  as  much  as  its  double  falls  short 
of  40.  Lets  =  the  number.  3x  -  50  =  40  -  2x;  5x  «  90;  a;  =  18,  ans. 
Proving,  54  -  50  =  40  -  36. 

Equations  containing  two  unknown  quantities.  —  If  one  equation 
contains  two  unknown  quantities,  x  and  y,  an  indefinite  number  of  pairs 
of  values  of  x  and  y  may  be  found  that  will  satisfy  the  equation,  but  if  a 
second  equation  be  given  only  one  pair  of  values  can  be  found  that  will 
satisfy  both  equations.  Simultaneous  equations,  or  those  that  may  be 
satisfied  by  the  same  values  of  the  unknown  quantities,  are  solved  by 
combining  the  equations  so  as  to  obtain  a  single  equation  containing  only 
one  unknown  quantity.  This  process  is  called  elimination. 

Elimination  by  addition  or  subtraction.  —  Multiply  the  equation  by 
such  numbers  as  will  make  the  coefficients  of  one  of  the  unknown  quanti- 
ties equal  in  the  resulting  equation.  Add  or  subtract  the  resulting  equa- 
tions according  as  they  have  unlike  or  like  signs. 

M 

Substituting  value  of  ?/  in  first  equation,  2x  4-  3  =  7;  x  =  2. 

Elimination  by  substitution.  —  From  one  of  the  equations  obtain  the 
value  of  one  of  the  unknown  quantities  in  terms  of  the  other.  Substi- 
tute for  this  unknown  quantity  its  value  in  the  other  equation  and  reduce 
the  resulting  equations. 


4  3y  ^  7.        Multiply  by  2:  .    _„ 

-  by  =  3.       Subtract :  4a?  —  5y  —  3    l\y  -  11 ;  y  =>  1« 


c^irr^  f  2.r  +  3y  =  8.     (1).       From  (1)  we  find  x 
bolvel3z  +7y  =  7.     (2). 

Substitute  this  value  in  (2):  3  (8  ~  3?/)  4-7^  =  7;  = 


whence  y  =  -  2.     Substitute  this  value  in  (1):  2x  —  6  =  8;  x  =  7. 

Elimination  by  comparison.  —  From  each  equation  obtain  the  value  of 
one  of  the  unknown  quantities  in  terms  of  the  9ther.  Form  an  equation 
from  these  equal  values,  and  reduce  this  equation. 

Solve  2x  —  9y  =  11.    (1)    and  3x  -  4y  =  7.     (2).     From  (1)  we  find 


From  (2)  we  find  x 


Equating  these  values  of  x, ll  t  9^  =  7  ~t  4y ;  IQy  -  -  19;  y  =  -  1. 

»j  O 

Substitute  this  value  of  y  in  (1):  2x  4-9  =  11;  x  =  1. 

If  three  simultaneous  equations  are  given  containing  three  unknown 
quantities,  one  of  the  unknown  quantities  must  be  eliminated  between  two 
pairs  of  the  equations;  then  a  second  between  the  two  resulting  equations. 

Quadratic  equations.  —  A  quadratic  equation  contains  the  square  of 
the  unknown  quantity,  but  no  higher  power.  A  pure  quadratic  contains 
the  square  only;  an  affected  quadratic  both  the  square  and  the  first  power. 

To  solve  a  pure  quadratic,  collect  the  unknown  quantities  on  one  side, 

id  the  known  quantities  on  the  other;  divide  by  the  coefficient  of  the 

iknown  quantity  and  extract  the  square  root  of  each  side  of  the  resulting 

[nation. 

Solve  3z2  -  15  =  0.     3z*  =  15;  x*  =  5;  x  =  >/5. 

A  root  like  x/5,  which  is  indicated,  but  which  can  be  found  only  approxi- 
'  Ay.  is  called  a  surd. 


36  ALGEBRA. 

Solve  3a*  +  15  -  0.     3z?=  -  15;  a*  =  -  5;  x  =  v. 

The  square  root  of  —  5  cannot  be  found  even  approximately,  fo/  tha 
square  of  any  number  positive  or  negative  is  positive;  therefore  a  root 
which  is  indicated,  but  cannot  be  found  even  approximately,  is  called 
imaginary. 

To  solve  an  affected  quadratic,  1.  Convert  the  equation  into  the  form 
a*x2  ±  2abx  =  c,  multiplying  or  dividing  the  equation  if  necessary,  so  as 
to  make  the  coefficient  of  x2  a  square  number. 

2.  Complete  the  square  of  the  first  member  of  the  equation,  so  as  to 
convert  it  to  the  form  of  a2x2  ±  2abx  +  b2,  which  is  the  square  of  the 
binomial  ax  ±  &,  as  follows:  add  to  each  side  of  the  equation  the  square  of 
the  quotient  obtained  by  dividing  the  second  term  by  twice  the  square 
root  of  the  first  term. 

3.  Extract  the  square  root  of  each  side  of  the  resulting  equation. 
Solve  3.*2  -  4.r  =  32.     To  make  the  coefficient  of  x2  a  square  number, 

multiply  by  3  :  9x2  -  I2x  =  96;  I2x  +  (2  X  3x)  =  2;  22  =  4. 

Complete  the  square:  9x2  —  I2x  +  4  =  100.  Extract  the  root: 
3x  —  2  =•  ±10,  whence  x  =  4  or  —  22/3.  The  square  root  of  100  is 
either  +  10  or  —  10,  since  the  square  of  —  10  as  well  as  +  102  =  100. 

Every  affected  quadratic  may  be  reduced  to  the  form  ax*+bx+c=-Q. 

The  solution  of  this  equation  is  x  =  --  -  — 

Problems  involving  quadratic  equations  have  apparently  two  solutions, 
as  a  quadratic  has  two  roots.  Sometimes  both  will  be  true  solutions,  but 
generally  one  only  will  be  a  solution  and  the  other  be  inconsistent  with  the 
conditions  of  the  problem. 

The  sum  of  the  squares  of  two  consecutive  positive  numbers  is  481. 
Find  the  numbers. 

Let  x  =.  one  number,  x+1  the  other.  z2  +  (x  -f  I)2  =  481.  2x*  -f 
2x  +  1  =  481. 

x2  +  x  =  240.  Completing  the  square,  x2  +x  -f  0.25  =  240.25. 
Extracting  the  root  we  obtain  x  +  0.5  =  ±  15.5;  x  =  15  or  -  16.  The 
negative  root  —  16  is  inconsistent  with  the  conditions  of  the  problem. 

Quadratic  equations  containing  two  unknown  quantities  require 
different  methods  for  their  solution,  according  to  the  form  of  the  equations. 
For  these  methods  reference  must  be  made  to  works  on  algebra. 

Theory  of  exponents.  —  %a  when  n  is  a  positive  integer  is  one  of  n 


equal  factors  of  a.      \o™  means  a  is  to  be  raised  to  the  with  power  and  the 
nth  root  extracted. 


tnat  the  nth  root  of  a  is  to  be  taken  and  the  result 
raised  to  the  with  power. 

«\/a™  =  (  \l~a\m  =  an.  When  the  exponent  is  a  fraction,  the  numera- 

tor indicates  a  power,  and  the  denominator  a  root.     a6/2  =  v/a6  =  a3; 
a3/2  =  Va3  =  a1-  s. 

To  extract  the  root  of  a  quantity  raised  to  an  indicated  power,  divide 
the  exponent  by  the  index  of  the  required  root;  as, 


Subtracting  1  from  the  exponent  of  a  is  equivalent  to  dividing  by  a: 
02-i=  a'  =o;  a'-i  =  a«  -  ^-  1;  a°-i  =  a~>  ~  i;  a-»-i=a-2=l. 

A  number  with  a  negative  exponent  denotes  the  reciprocal  of  the  num- 
ber with  the  corresponding  positive  exponent. 

A  factor  under  the  radical  sign  whose  root  can  be  taken  may,  by  having 
the  root  taken,  be  removed  from  under  the  radical  sign: 


GEOMETRICAL   PROBLEMS. 


37 


A  factor  outside  the  radical  sign  may  be  raised  to  the  corresponding 
power  and  placed  under  it: 


Binomial  Theorem. 

sion  of  the  form  x  +  a 


-  To  obtain  any  power,  as  the  nth,  of  an  expres- 


•x*  + 


etc.  *~*  i-2-3- 

The  following  laws  hold  for  any  term  in  the  expansion  of  (a  4-  x)n. 

The  exponent  of  x  is  less  by  one  than  the  number  of  terms. 

The  exponent  of  a  is  n  minus  the  exponent  of  x. 

The  last  factor  of  the  numerator  is  greater  by  one  than  the  exponent  of  a. 

The  last  factor  of  the  denominator  is  the  same  as  the  exponent  of  x. 

In  the  rth  term  the  exponent  of  x  will  be  r  —  1. 

The  exponent  of  a  will  be  n  —  (r  —  1),  or  n  —  r  4-  1. 

The  last  factor  of  the  numerator  will  be  n  —  r  4-  2. 

The  last  factor  of  the  denominator  will  be  =  r  —  1. 

Hence  the  rth  term  =  "(»  -  D(»  -  2)  .      («  -  r+ 2)  ^ 

l.^.O....^?*  —    1^ 


GEOMETRICAL  PROBLEMS. 


1.  To  bisect  a  straight  line,  or 
an  arc  of  a  circle  (Fig.  1).  —  From 
the  ends  A,  B,  as  centres,  describe 
arcs  intersecting  at  C  and  D,  and 
draw  a  line  through  C  and  D  which 
will  bisect  the  line  at  E  or  the  arc 
at  F. 

2.  To  draw  a  perpendicular  to 
a  straight  line,  or  a  radial  line  to 
a     circular     arc.  —  Same     as     in 
Problem  1.     C  D  is  perpendicular  to 
the  line  A  B,  and  also  radial  to  the 
arc. 

3.  To  draw  a  perpendicular  to 
a  straight  line  from  a  given  point 
in  that  line  (Fig.  2).  —  With  any 
radius,  from  the  given  point  A  in  the 
line  B  C,  cut  the  line  at  B  and  C. 
With  a  longer  radius  describe  arcs 
from  B  and  C,  cutting  each  other  at 
D,  and  draw  the  perpendicular  D  A. 

4.  From  the  end  A  of  a  given 
line  A  D  to  erect  a  perpendicular 
AE  (Fig.  3).  —  From  any  centre  F, 
above  A  D,  describe  a  circle  passing 
through  the  given  point  A ,  and  cut- 
ting the  given  line  at  D.     Draw  D  F 
and  produce  it  to  cut  the  circle  at  Et 
and  draw  the  perpendicular  A  E. 

Second  Method  (Fig.  4).  —  From 
the  given  point  A  set  off  a  distance 
A  E  equal  to  .three  parts,  by  any 
scale;  and  on  the  centres  A  and  E, 
with  radii  of  four  and  five  parts 
respectively,  describe  arcs  intersect- 
ing at  C,  Draw  the  perpendicular 
A  C. 

NOTE.  —  This  method  is  most 
useful  on  very  large  scales,  where 
straight  edges  are  inapplicable.  Any 
multiples  of  the  numbers  3,  4,  5  may 
be  taken  with  the  same  effect,  as  6,  & 
10,  or  9,  12.  15. 


38 


GEOMETRICAL   PROBLEMS. 


5.  To  draw  a  perpendicular  to 
a  straight  line  from  any  point 
without  it  (Fig.  5).  —  From  the 
point  A,  with  a  sufficient  radius  cut 
the  given  line  at  F  and  G,  and  from 
these  points  describe  arcs  cutting  at 
E.  Draw  the  perpendicular  A  E. 


6.  To  draw  a  straight  line 
parallel  to  a  given  line,  at  a  given 
distance  apart  (Fig.  6).  —  From 
the  centres  A,  B,  in  the  given  line, 
with  the  given  distance  as  radius, 
describe  arcs  (7,  D,  and  draw  the 
parallel  lines  C  D  touching  the  arcs. 


7.  To  divide  a  straight  line  into 
a  number  of  equal  parts  (Fig.  7). 
—  To  divide  the  line  A  B  into,  say, 
five  parts,  draw  the  line  A  C  at  an 
angle  from  A ;  set  off  five  equal  parts; 
draw  B5  and  draw  parallels  to  it 
from  the  other  points  of  division  in 
A  C.  These  parallels  divide  A  B  as 
required. 

NOTE.  —  By  a  similar  process  a 
line  may  be  divided  into  a  number 
of  unequal  parts;  setting  off  divisions 
on  A  C,  proportional  by  a  scale  to  the 
required  divisions,  and  drawing 
parallels  cutting  A  B.  The  triangles 
All,  A 22,  A33,  etc.,  are  similar 
triangles. 


8.   Upon  a  straight  line  to  draw 
an  angle  equal  to  a  given  angle 

(Fig.  8).  —  Let  A  be  the  given  angle 
and  F  G  the  line.  From  the  point  A 
with  any  radius  describe  the  arc  D  E. 
From  F  with  the  same  radius 
describe  I  H.  Set  off  the  arc  I H 
equal  to  D  E,  and  draw  F  H.  The 
angle  F  is  equal  to  A,  as  required. 


9.   To  draw  angles  of  60°  and 

80°  (Fig.  9).  —  From  F,  with  any 
radius  F  /,  describe  an  arc  /  H ;  and 
from  /,  with  the  same  radius,  cut 
the  arc  at  H  and  draw  F  H  to  form 
the  required  angle  I  F  H.  Draw  the 
perpendicular  H  K  to  the  base  line  to 
form  the  angle  of  30°  F  H  K. 


10.   To   draw  an  angle   of  45° 

(Fig.  10).  —  Set  off  the  distance  F  /; 
draw  the  perpendicular  /  H  equal  to 
/  Ft  and  join  H  " 


\. 


FIG.  9. 


F. 


Fto  form  the  angle  at 
The  angle  at  H  is  "also  45°. 


FIG.  10. 


GEOMETRICAL   PROBLEMS. 


39 


FIG.  11. 


Fia.  15. 


11.  To  bisect  an  angle  (Fig.  11). 
—  Let  ACB  be  the  angle;  with  C  as 
a  centre  draw  an  arc  cutting  the 
sides  at  A,  B.     From  A  and  B  as 
centres,   describe  arcs  cutting  each 
other  at  Z>.     Draw  C  D,  dividing  the 
angle  into  two  equal  parts. 

12.  Through  two  given  points 
to  describe  an  arc  of  a  circle  with 
a  given  radius   (Fig.  12).  —  From 
the  ppints  A  and  B  as  centres,  with 
the  given  radius,  describe  arcs  cut- 
ting at  C;  and  from  C  with  the  same 
radius  describe  an  arc  A  B. 

13.  To  find  the  centre  of  a  circle 
or  of  an  arc  of  a  circle  (Fig.  13).  — 
Select  three  points,  A,  B,  C,  in  the 
circumference,  well  apart;  with  the 
same  radius  describe  arcs  from  these 
three  points,  cutting  each  other,  and 
draw    the    two    lines,     D  E,     FG, 
through     their     intersections.     The 
point  O,  where  they  cut,  is  the  centre 
of  the  circle  or  arc. 

To  describe  a  circle  passing 
through  three  given  points.  — 
Let  A,  B,  C  be  the  given  points,  and 
proceed  as  in  last  problem  to  find  the 
centre  O,  from  which  the  circle  may 
be  described. 


14.  To  describe  an  arc  of  a 
circle  passing  through  three 
given  points  when  the  centre  is 
not  available  (Fig.  14).  —  From 
the  extreme  points  A,  B,  as 
centres,  describe  arcs  AH,  B  G. 
Through  the  third  point  C  draw 
A  E:  B  F,  cutting  the  arcs. 
Divide  A  F  and  B  E  into  any 
number  of  equal  parts,  and  set 
off  a  series  of  equal  parts  of  the 
same  length  on  the  upper  por- 
tions of  the  arcs  beyond  the 
points  E  F.  Draw  straight 
lines,  B  L,  BM,  etc.,  to  the 
divisions  in  A  F,  and  A  I,  A  K, 
etc.,  to  the  divisions  in  EG. 
The  successive  intersections  N, 
O,  etc.,  of  these  lines  are  points 
in  the  circle  required  between  the 
given  points  A  and  C,  which  may 
be  drawn  in;  similarly  the  remain- 
ing part  of  the  curve  BC  may 
be  described.  (See  also  Problem 
54.) 


15.  To  draw  a  tangent  to  a 
circle  from  a  given  point  in  the 
circumference  (Fig.  15). —  Through 
the  given  point  A,  draw  the  radial 
line  A  C,  and  a  perpendicular  to  it, 
FGt  which  is  "the  tangent  required. 


40 


GEOMETRICAL  PROBLEMS. 


16.  To    draw    tangents    to    a 
circle  from  a  point  without  it  (Fig. 
16).  —  From    A,    with    the    radius 
A  C,    describe   an    arc    BCD,    and 
from  C,  with  a  radius  equal  to  the 
diameter  of  the  circle,  cut  the  arc  at 
BD.      Join   BC,   CD,   cutting   the 
circle  at  E  F,  and  draw  A  E,  AF, 
the  tangents. 

NOTE.  —  When  a  tangent  is 
already  drawn,  the  exact  point  of 
contact  may  be  found  by  drawing  a 
perpendicular  to  it  from  the  centre. 

17.  Between  two  inclined  lines 
to  draw  a  series  of  circles  touching 
these    lines    and    touching    each 
other  (Fig.  17).  —  Bisect  the  inclina- 
tion of  the  given  lines  A  B,  C  D,  by 
the  line  N  O.    From  a  point  P  in  this 
line  draw  the  perpendicular  P  B  to  the 
line  A  B,  and  on  P  describe  the  circle 
B  D,  touching  the  lines  and  cutting 
the  centre  line  at  E.     From  E  draw 
E  F  perpendicular  to  the  centre  line, 
cutting    A  B    at    F,    and    from    F 
describe  an  arc  E  G,  cutting  A  B  at 
G.     Draw     GH    parallel     to     B  P, 
giving   H,   the  centre  of  the  next 
circle,    to    be    described    with    the 
radius  HE,  and  so  on  for  the  next 
circle  IN. 

Inversely,  the  largest  circle  may 
be  described  first,  and  the  smaller 
ones  in  succession.  This  problem  is 
of  frequent  use  in  scroll-work. 

18.  Between  two  inclined  lines 
to  draw  a  circular  segment  tan- 
gent   to    the    lines    and    passing 
through  a  point  F  on  the  line  FC 
which   bisects    the   angle   of   the 
lines  (Fig.  18).  — Through  F  draw 
DA    at  right  angles  to  FC;  bisect 
the  angles  A  and  Z),  as  in  Problem 
11,  by  lines  cutting  at  C,  and  from 
C  with  radius  C  F  draw  the  arc  H  F  G 
required. 

19.  To  draw  a  circular  arc  that 
will  be  tangent  to  two  given  lines 
AB  and  C  D  inclined  to  one  another, 
one  tangential  point  E  being  given 

(Fig.  19).  —  Draw  the  centre  line 
GF.  From  E  draw  E  F  at  right 
angles  to  A  B ;  then  F  is  the  centre 
of  the  circle  required. 

20.  To  describe  a  circular  arc 
Joining  two  circles,  and  touching 
one  of  them  at  a  given  point  (Fig. 
20).  —  To  join  the  circles  A  B,  FG, 
by  an  arc  touching  one  of  them  at 
F,  draw  the  radius  E  F,  and  produce 
it  both  ways.     Set  off  F  H  equal  to 
the  radius  A  C  of  the  other  circle; 
join  CH  and  bisect  it  with  the  per- 
pendicular   L  I,    cutting  E  F  at   I. 
On  the  centre  7,   with   radius  IF, 
describe  the  arc  FA  as  required. 


GEOMETRICAL   PROBLEMS. 


FIG.  22. 
E 


FIG.  23. 


FIG.  24. 


FIG.  25. 


FIG.  26. 


21.  To  draw  a  circle  with  a 
given  radius  R  that  will  be  tan- 
gent to  two  given  circles  A  and  B 

(Fig.  21).  —  From  centre  of  circle 
A  with  radius  equal  R  plus  radius 
of  A,  and  from  centre  of  B  with 
radius  equal  to  R  +  radius  of  B, 
draw  two  arcs  cutting  each  other  in 
C,  which  will  be  the  centre  of  the 
circle  required. 


22.   To  construct  an  equilateral 
triangle,    the    sides    being    given 

(Fig.  22).  —  On  the  ends  of  one  side, 
A,  B,  with  A  B  as  radius,  describe 
arcs  cutting  at  C,  and  draw  A  C,  C  B. 


23.  To  construct  a  triangle  of 
unequal  sides  (Fig.  23).  —  On 
either  end  of  the  base  A  D,  with  the 
side  B  as  radius,  describe  an  arc; 
and  with  the  side  C  as  radius,  on  the 
other  end  of  the  base  as  a  centre,  cut 
the  arc  at  E.  Join  A  E,  D  E. 


24.  To  construct  a  square  on™ 
given  straight  line  A  B  (Fig.  24). 
—  With  A  B  as  radius  and  A  and  B 
as  centres,  draw  arcs  A  D  and  B  C, 
intersecting  at  E.  Bisect  E  B  at 
F.  With  E  as  centre  and  E  F  as 
radius,  cut  the  arcs  A  D  and  B  C 
in  D  and  C.  Join  A  C,  C  Dt  and 
D  B  to  form  the  square. 


25.   To    construct    a    rectangle 
with  given  base  E  F  and  height  EH 

(Fig.  25).  —  On  the  base  E  F  draw 
the  perpendiculars  E  //,  F  O  equal 
to  the  height,  and  join  G  H. 


26.  To  describe  a  circle  about 
a  triangle  (Fig.  26).  —  Bisect  two 
sides  A  B,  A  C  of  the  triangle  at 
E  F,   and   from  these  points  draw 
perpendiculars    cutting    at    K.     On 
the  centre  K,  with  the  radius  K  A, 
draw  the  circle  ABC. 

27.  To   inscribe  a   circle  in  a 
triangle  (Fig.  27).— Bisect  two  of 
the  angles  A,  C,  of  the  triangle  by 


42 


GEOMETRICAL    PROBLEMS. 


lines  cutting  at  D;  from  D  draw  a 
perpendicular  D  E  to  any  side,  and 
with  D  E  as  radius  describe  a  circle. 
When  the  triangle  is  equilateral, 
draw  a  perpendicular  from  one  of  the 
angles  to  the  opposite  side,  and  from 
the  side  set  off  one  third  of  the 
perpendicular. 

28.  To  describe  a  circle  about 
a  square,  and  to  inscribe  a  square 
in  a  circle  (Fig.  28).  —  To  describe 
the  circle,  draw  the  diagonals  A  B, 
C  D  of  the  square,  cutting  at  E.     On 
the  centre  E,  with  the  radius  A  E, 
describe  the  circle. 

To  inscribe  the  square.  —  Draw 
the  two  diameters,  A  B,C  D,  at  right 
angles,  and  join  the  points  A,  B, 
C  D,  to  form  the  square. 

NOTE.  —  In  the  same  way  a  circle 
may  be  described  about  a  rectangle. 

29.  To   inscribe   a   circle   in  a 
square  (Fig.  29).  —  To  inscribe  the 
circle,  draw  the  diagonals  A  B,  C  D 
of  the  square,  cutting  at  E;  draw  the 
perpendicular  E  F  to  one  side,  and 
with  the  radius  E  F  describe  the 
circle. 


FIG.  28. 
A      G  C 


30.  To  describe  a  square  about 
a  circle  (Fig.  30).  —  Draw  two 
diameters  A  B,  C  D  at  right  angles. 
With  the  radius  of  the  circle  and 
A,  B,  C  and  D  as  centres,  draw  the 
four  half  circles  which  cross  one 
another  in  the  corners  of  the  square. 


31.  To  inscribe  a  pentagon  in 
a  circle  (Fig.  31).  —  Draw  diam- 
eters A  C,  B  D  at  right  angles,  cut- 
ting at  o.  Bisect  A  o  at  E,  and  from 
E,  with  radius  E  B,  cut  A  C  at  F; 
from  B,  with  radius  B  F,  cut  the 
circumference  at  G,  H,  and  with  the 
same  radius  step  round  the  circle  to 
/  and  K\  join  the  points  so  found  to 
form  the  pentagon. 


32.  To  construct  a  pentagon 
on  a  given  line  A  B  (Fig.  32).— 
From  B  erect  a  perpendicular  B  C 
half  the  length  of  A  B ;  join  A  C  and 
prolong  it  to  D,  making  C  D  =  B  C. 
Then  B  D  is  the  radius  of  the  circle 
circumscribing  the  pentagon.  From 
A  and  B  as  centres,  with  B  D  as 
radius,  draw  arcs  cutting  each  other 
in  O,  which  is  the  centre  of  the  circle. 


FIG.  32. 


GEOMETRICAL    PROBLEMS. 


43 


FIG.  34. 


33.  To    construct    a    hexagon 
upon  a   given   straight  line   (Fig. 

33).  —  From  A  and  B,  the  ends  of 
the  given  line,  with  radius  A  B, 
describe  arcs  cutting  at  g;  from  gt 
with  the  radius  g  A,  describe  a  circle; 
with  the  same  radius  set  off  the  arcs 
A  G,  G  F,  and  B  D,  D  E.  Join  the 
points  so  found  to  form  the  hexagon. 
The  side  of  a  hexagon  =  radius  of  its 
circumscribed  circle. 

34.  To  inscribe  a  hexagon  in  a 
circle    (Fig.    34).  —  Draw   a   diam- 
eter  ACS.     From   A    and  B  as 
centres,  with  the  radius  of  the  circle 
A  C,  cut  the  circumference,  at  D,  E, 
F,  G,  and  draw  A  D,  D  E,  etc.,  to 
form  the  hexagon.     The  radius  of 
the  circle  is  equal  to  the  side  of  the 
hexagon;  therefore  the  points  D,  Et 
etc.,  may  also  be  found  by  stepping 
the  radius  six  times  round  the  circle. 
The  angle  between  the  diameter  and 
the  sides  of  a  hexagon  and  also  the 
exterior  angle  between  a  side  and  an 
adjacent  side  prolonged  is  60  degrees; 
therefore  a  hexagon    may    conven- 
iently be  drawn  by  the  use  of  a  60- 
degree  triangle. 

35.  To     describe     a     hexagon 
about  a  circle  (Fig.  35).  —  Draw  a 
diameter  A  D  B,  and  with  the  radius 
A  D,  on  the  centre  A,  cut  the  circum- 
ference at  C;  join  A  C,  and  bisect  it 
with  the  radius  D  E ;  through  E  draw 
FG,  parallel  to  A  C,  cutting  the  diam- 
eter at  F,  and  with  the  radius  D  F 
describe    the    circumscribing    circle 
F  H.     Within  this  circle  describe  a 
hexagon  by  the  preceding  problem. 
A  more  convenient  method  is  by  use 
of  a  60-degree  triangle.     Four  of  the 
sides  make  angles  of  60  degrees  with 
the  diameter,  and  the  other  two  are 
parallel  to  the  diameter. 

36.  To  describe  an  octagon  on 
a  given  straight  line  (Fig.  36).  — 
Produce   the  given  line  'A  B  both 
ways,  and  draw  perpendiculars  A  E. 
BF;  bisect  the  external  angles^,  and 
B  by  the  lines  A  H,  B  C,  which  make 
equal  to  A  B.     Draw  C  D  and  H  G 
parallel  to  A  E,  and  equal  to  A  B; 
from   the   centres   G,    D,    with   the 
radius  A  B,  cut  the  perpendiculars  at 
E,  F,  and  draw  E  F  to  complete  the 
octagon. 

37.  To   convert  a   sqaare   into 
an  octagon  (Fig.  37).  —  Draw  the 
diagonals  of  the  square  cutting  at  e; 
from  the  corners  A,  B,  C,  D,  with 
A  e  as  radius,  describe  arcs  cutting 
the  sides  at  gn,  fk,  hm,  and  ol,  and 
join  the  points  so  found  to  form  the 
octagon.     Adjacent  sides  of  an  octa- 
gon make  an  angle  of  135  degrees. 


GEOMETRICAL  PROBLEMS. 


38.  To  inscribe  an  octagon  in 
a  circle  (Fig.  38).  —  Draw  two 
diameters,  A  C,  B  D  at  right  angles; 
bisect  the  arcs  A  B,  B  C,  etc.,  at  e  f, 
etc.,  and  join  A  e,  €  B,  etc.,  to  form 
the  octagon, 


39.  To  describe  an  octagon 
about  a  circle  (Fig.  39).  —  P -scribe 
a  square  about  the  given  circle  A  B; 
draw  perpendiculars  h  k,  etc.,  to  the 
diagonals,  touching  the  circle  to 
form  the  octagon. 


40.  To  describe  a  polygon  of 
any  number  of  sides  upon  a  given 
straight  line  (Fig.  40).  —  Produce 
the  given  .line  A  B,  and  on  A,  with  the 
radius  A  B,  describe  a  semicircle; 
divide  the  semi-circumference  into 
as  many  equal  parts  as  there  are  to 
be  sides  in  the  polygon  —  say,  in 
this  example,  five  sides.  Draw  lines 
from  A  through  the  divisional  points 
D,  b,  and  c,  omitting  one  point  a; 
and  on  the  centres  B,  D,  with  the 
radius  A  B,  cut  A  b  at  E  and  A  c  at  F. 
Draw  D  E,  E  Ft  F  B  to  complete  the 
polygon. 


41.  To  inscribe  a  circle  within 
a  polygon  (Figs.  41,  42).  —  When 
the  polygon  has  an  even  number  of 
sides  (Fig.  41),  bisect  two  opposite 
sides  at  A  and  B;  draw  A  B,  and 
bisect  it  at  C  by  a  diagonal  D  E,  and 
with  the  radius  C  A  describe  the 
circle. 

When  the  number  of  sides  is  odd 
(Fig.  42),  bisect  two  of  the  sides  at  A 
and  B,  and  draw  lines  A  E,  B  D  to  the 
opposite  angles,  intersecting  at  C; 
from  <7,  with  the  radius  C  A,  describe 
the  circle. 


42.  To  describe  a  circle  without 
a  polygon   (Figs.   41,   42).  —  Find 
the  centre  C  as  before,  and  with  the 
radius  C  D  describe  the  circle. 

43.  To   inscribe   a   polygon   of 
any  number  of  sides  within  a  circle 

(Fig.  43).  —Draw  the  diameter  A  B 
and  through  the  centre  E  draw  the 


H  D  G 

FIG.  39. 


Fio.  42. 


GEOMETRICAL   PROBLEMS. 


45 


perpendicular  E  C,  cutting  the  circle 
at  F.  Divide  E  F  into  four  equal 
parts,  and  set  off  three  parts  equal 
to  those  from  F  to  C.  Divide  the 
diameter  A  B  into  as  many  equal 
parts  as  the  polygon  is  to  have  sides; 
and  from  C  draw  C  D,  through  the 
second  point  of  division,  cutting  the 
circle  at  D.  Then  A  D  is  equal  to  one 
side  of  the  polygon,  and  by  stepping 
round  the  circumference  with  the 
length  A  D  the  polygon  may  be  com- 
pleted. 


Table  of  Polygonal  Angles. 


Number 
of  Sides. 

Angle 
at  Centre. 

Number 
of  Sides. 

Angle 
at  Centre. 

Number 
of  Sides. 

Angle 
at  Centre. 

No. 

4 
5 
6 

8 

Degrees. 
120 
90 
72 
60 

g* 

No. 
9 
10 
11 
12 
13 
14 

Degrees. 
40 
36 

gw 
i£ 

No. 
15 
16 
17 
18 
19 
20 

Degrees. 
22l/2 

iH 

19 
18 

In  this  table  the  angle  at  the  centre  is  found  by  dividing  360  degrees,  the 
number  of  degrees  in  a  circle,  by  the  number  of  sides  in  the  polygon;  and 
by  setting  off  round  the  centre  of  the  circle  a  succession  of  angles  by  means 
of  the  protractor,  equal  to  the  angle  in  the  table  due  to  a  given  number  of 
sides,  the  radii  so  drawn  will  divide  the  circumference  into  the  same  num- 
ber of  parts. 

44.  To  describe  an  ellipse  when 
the  length  and  breadth  are  given 
(Fig.  44). — A  B,  transverse  axis; 
C  Z>,  conjugate  axis;  F  G,  foci.  The 
sum  of  the  distances  from  C  to  F 
and  G,  also  the  sum  of  the  distances 
from  F  and  G  to  any  other  point  in 
the  curve,  is  equal  to  the  transverse 
axis.  From  the  centre  C,  with  A  E 
as  radius,  cut  the  axis  A  B  at  F  and 
G,  the  foci;  fix  a  couple  of  pins  into 
the  axis  at  F  and  G,  and  loop  on  a 
thread  or  cord  upon  them  equal  in 
length  to  the  axis  A  B,  so  as  when 
stretched  to  reach  to  the  extremity 
C  of  the  conjugate  axis,  as  shown  in 
dot-lining.  Place  a  pencil  inside  the 
cord  as  at  //,  and  guiding  the  pencil 
in  this  way,  keeping  the  cord  equally 
in  tension,  carry  the  pencil  round  the 
pins  F,  G,  and  so  describe  the 
ellipse. 

NOTE.  —  This  method  is  employed 
in  setting  off  elliptical  garden-plots, 
walks,  etc. 

2d  Method  (Fig.  45).  —  Along  the 
straight  edge  of  a  slip  of  stiff  paper 
mark  off  a  distance  a  c  equal  to  A  C, 
half  the  transverse  axis;  and  from 
the  same  point  a  distance  a  b  equal 
to  C  G,  half  the  conjugate  axis. 


FIG.  44. 


GEOMETRICAL  PROBLEMS. 


Place  the  slip  so  as  to  bring  the  point  b  on  the  line  A  B  of  the  transverse 
axis,  and  the  ppint  c  on  the  line  D  E;  and  set  off  on  the  drawing  the  posi- 
tion of  the  point  a.  Shifting  the  slip  so  that  the  point  b  travels  on  the 
transverse  axis,  and  thexpoint  c  on  the  conjugate  axis,  any  number  of 
points  in  the  curve  may  be  found,  through  which  the  curve  may  be 
traced. 

3d  Method  (Fig.  46).  —  The  action 
of  the  preceding  method  may  be 
embodied  so  as  to  afford  the  means 
of  describing  a  large  curve  contin- 
uously by  means  of  a  bar  m  k,  with 
steel  points  m,  I,  k,  riveted  into  brass 
slides  adjusted  to  the  length  of  the 
semi-axis  and  fixed  with  set-screws. 
A  rectangular  cross  E  G,  with  guiding- 
slots  is  placed,  coinciding  with  the 
two  axes  of  the  ellipse  A  C  and  B  H. 
B7  sliding  the  points  k,  I  in  the  slots, 
and  carrying  round  the  point  m,  the 
curve  may  be  continuously  described. 
A  pen  or  pencil  may  be  fixed  at  m. 

4th  Method  (Fig.  47).  —  Bisect  the 
transverse  axis  at  C,  and  through  C  * 
draw  the  perpendicular  D  E,  making 
C  D  and  C  E  each  equal  to  half  the 
conjugate  axis.  From  D  or  E,  with 
the  radius  AC,  cut  the  transverse 
axis  at  F,  Ff,  for  the  foci.  Divide 
A  C  into  a  number  of  parts  at  the 
points  1,  2,  3,  etc.  With  the  radius 
Al  on.  F  and  F'  as  centres,  describe 
arcs,  and  with  the  radius  B  1  on  the 
same  centres  cut  these  arcs  as  shown. 
Repeat  the  operation  for  the  other 
divisions  of  the  transverse  axis.  The 
series  of  intersections  thus  made  are 
points  in  the  curve,  through  which 
the  curve  may  be  traced. 

5th  Method  (Fig.  48).  —  On  the 
two  axes  A  B,  D  E  as  diameters,  on 
centre  C,  describe  circles;  from  a 
number  of  points  a,  b,  etc.,  in  the 
circumference  A  F  B,  draw  radii  cut- 
ting the  inner  circle  at  a',  6',  etc. 
From  a,  b,  etc.,  draw  perpendiculars 
to  AB;  and  from  a',  b't  etc.,  draw 
parallels  to  A  B,  cutting  the  respec- 
tive perpendiculars  at  n,  o,  etc.  The 
intersections  are  points  in  the  curve, 
through  which  the  curve  may  be 
traced. 

6to  Method  (Fig.  49).  —  When  the 
transverse  and  conjugate  diameters 
are  given,  A  B,  CD,  draw  the  tangent 
EF  parallel  to  A  B.  Produce  CD, 
and  on  the  centre  G  with  the  radius 
of  half  A  B,  describe  a  semicircle 
H  D  K;  from  the  centre  G  draw  any 
number  of  straight  lines  to  the  points 
E,  r,  etc.,  in  the  line  E  Ft  cutting  the 
circumference  at  /,  m,  n,  etc.;  from 
the  centre  O  of  the  ellipse  draw 
straight  lines  to  the  points  E,  r,  etc.; 
and  from  the  points  I,  m,  n,  etc., 
draw  parallels  to  G  C,  cutting  the 
tines  O  E,  O  rt  etc.,  at  Lt  Mt  3v,  etc.  Fio.  49. 


GEOMETRICAL    PROBLEMS. 


47 


These  are  points  in  the  circumference  of  the  ellipse,  and  the  curve  may  be 
traced  through  them.  Points  in  the  other  half  of  the  ellipse  are  formed 
by  extending  the  intersecting  lines  as  indicated  in  the  figure. 

45.  To  describe  an  ellipse 
approximately  by  means  of  cir- 
cular arcs.  —  First.  —  With  arcs 
of  two  radii  (Fig.  50).  —  Find  the 
difference  of  the  semi-axes,  and  set 
it  off  from  the  centre  O  to  a  and  c  on 
O  A  and  O  C;  draw  ac,  and  set  off 
half  a  c  to  d;  draw  d  i  parallel  to  a  c; 
set  off  O  e  equal  to  O  d;  join  e  i,  and 
draw  the  parallels  e  m,  d  m.  From 
m,  with  radius  m  C,  describe  an  arc 
through  C;  and  from  i  describe  an 
arc  through  D ;  from  d  and  e  describe 
arcs  through  A  and  B.  The  four 
arcs  form  the  ellipse  approximately. 
NOTE.  —  This  method  does  not 
apply  satisfactorily  when  the  con- 
jugate axis  is  less  than  two  thirds  of 
the  transverse  axis. 

2d  Method  (by  Carl  G.  Barth,  Fig. 
51).  —  In  Fig.  51  a  b  is  the  major  and 
c  d  the  minor  axis  of  the  ellipse  to  be 
approximated.  Lay  off  b  e  equal  to 
the  semi-minor  axis  c  O,  and  use  a  e 
as  radius  for  the  arc  at  each  extrem- 
ity of  the  minor  axis.  Bisect  e  o  at  / 
and  lay  off  eg  equal  toef,  and  use  gb 
as  radius  for  the  arc  at  each  extrem- 
ity of  the  major  axis. 

method  is  not  considered  applicable  for  cases  in  which  the  minor 
less  than  two  thirds  of  the  major. 

3d  Method:  With  arcs  of  three  radii 
(Fig.  52).  —  On  the  transverse  axis 
A  B  draw  the  rectangle  B  G  on  the 
height  O  C;  to  the  diagonal  A  C 
draw  the  perpendicular  G  H  D;  set 
off  O  K  equal  to  O  C,  and  describe  & 
semicircle  on  A  K,  and  produce  O  C 
to  L;  set  off  O  M  equal  to  C  L,  and 
from  D  describe  an  arc  with  radius 
D  M]  from  A,  with  radius  O  L,  cut 
A  B  at  N;  from  H,  with  radius  HN, 
cut  arc  a  b  at  a.  Thus  the  five 
centres  D,  a,  b,  H,  Hf  are  found, 
from  which  the  arcs  are  described  to 
form  the  ellipse. 

This  process  works  well  for  nearly 
all  proportions  of  ellipses.  It  is  used 
in  striking  out  vaults  and  stone 
bridges. 

4th  Method  (by  F.  R.  Honey, 
Figs.  53  and  54).  —  Three 
radii  are  employed.  With 
the  shortest  radius  describe 
the  two  arcs  which  pass 
through  the  vertices  of  the 
major  axis,  with  the  longest 
the  two  arcs  which  pass 

through  the  vertices  of  the 

minor  axis,  and  with  the  third 
radius  the  four  arcs  which 
connect  the  former. 


The 
axis  is 


b        Jid 

FIG.  53. 


48 


GEOMETRICAL   PROBLEMS. 


A  simple  method  of  determining  the  radii  of  curvature  is  illustrated  in 
Fig.  53.  Draw  the  straight  lines  a  f  and  a  c,  forming  any  angle  at  a.  With 
a  as  a  centre,  and  with  radii  a  b  and  a  c,  respectively,  equal  to  the  semi- 
minor  and  semi-major  axes,  draw  the  arcs  b  e  and  c  d.  Join  e  d,  and 
through  b  and  c  respectively  draw  b  g  and  c  f  parallel  to  e  d,  intersecting 
a  c  at  g,  and  a  /  at  /;  a  f  is  the  radius  of  curvature  at  the  vertex  of 
the  minor  axis;  and  a  g  the  radius  of  curvature  at  the  vertex  of  the 
major  axis. 

Lay  off  d  h  (Fig.  53)  equal  to  one  eighth  of  6  d.  Join  e  h,  and  draw  c  k 
and  b  I  parallel  to  e  h.  Take  a  k  for  the  longest  radius  ( =  R),  a  I  for  the 
shortest  radius  (=  r),  and  the  arithmetical  mean,  or  one  half  the  sum  of 
the  semi-axes,  for  the  third  radius  (=  p),  and  employ  these  radii  for  the 
eight-centred  oval  as  follows: 

Let  a  Sander/  (Fig.  54) 
be  the  major  and  minor 
axes.  Lay  off  a  e  equal 
to  r,  and  a  f  equal  to  p; 
also  lay  off  c  g  equal  to  R, 
and  c  h  equal  to  p'.  With 
g  as  a  centre  and  gfi  as  a 
radius,  draw  the  arc  h  k; 
with  the  centre  e  and 
radius  e  f  draw  the  arc  /  k,  a 
intersecting  h  k  at  k. 
Draw  the  line  g  k  and 


produce   it,    making   g   I 
equal   to    R.     Draw   k   e 


and  produce  it,  making 
k  m  equal  to  p.  With  the 
centre  g  and  radius  g  c 
(=  R)  draw  the  arc  c  I; 
with  the  centre  k  and 
radius  kl  (=  p)  draw  the 
arc  I  m,  and  with  the 
centre  e  and  radius  e  m 
(=  r)  draw  the  arc  m  a. 

The  remainder  of  the  work  is  symmetrical  with  respect  «,o  the 
axes. 

46.  The  Parabola.  —  A  parabola  (D  A  C,  Fig.  55)  is  a  curve  such 
that  every  point  in  the  curve  is  equally  distant  from  the  directrix  K  L 
and  the  focus  F.  The  focus  lies  in  the  axis 

A  B  drawn  from  the  vertex  or  head  of  the       K  P  \ 

curve  A,  so  as  to  divide  the  figure  into  two 
equal  parts.  The  vertex  A  is  equidistant 
from  the  directrix  and  the  focus,  or  A  e  =  A  F. 
Any  line  parallel  to  the  axis  is  a  diameter. 
A  straight  line,  as  E  G  or  DC,  drawn  across 
the  figure  at  right  angles  to  the  axis  is  a 
double  ordinate,  and  either  half  of  it  is  an 
ordinp.te.  The  ordinate  to  the  axis  E  F  G, 
drawn  through  the  focus,  is  called  the  para- 
meter of  the  axis.  A  segment  of  the  axis, 
reckoned  from  the  vertex,  is  an  abscissa  of 
the  axis,  and  it  is  an  abscissa  of  the  ordinate 
drawn  from  the  base  of  the  abscissa.  Thus, 
A  B  is  an  abscissa  of  the  ordinate  B  C. 


E 
«/ 

A 

L 

/^ 

\^\ 

F 

\ 

n/ 

O 

\ 

o 

\ 

\ 

T 

o 

D                  B 

b 

^-a      C 

FIG.  55. 


Abscissae  of  a  parabola  are  as  the  squares  of  their  ordinates. 


To  describe  a  parabola  when  an  abscissa  and  its  ordinate  are  given 

(Fig.  55).  —  Bisect  the  given  ordinate  B  C  at  a,  draw  A  a,  and  then  a  b 
perpendicular  to  it,  meeting  the  axis  at  6.  Set  off  A  e,  A  F,  each  equal  to 
B  b;  and  draw  K  e  L  perpendicular  to  the  axis.  Then  K  L  is  the  directrix 
and  F  is  the  focus.  Through  F  and  any  number  of  points,  o,  o,  etc.,  in  the 
axis,  draw  double  ordinates,  n  o  n,  etc.,  and  from  the  centre  F,  with  the 
radii  F  et  o  e,  etc.,  cut  the  respective  ordinates  at  E,  G,  n,  n,  etc..  The 
curve  may  be  traced  through  these  points  as  shown. 

2d  Method:  By  means  of  a  square  and  a  cord  (Fig.  56).  —  Place  a 


GEOMETRICAL   PROBLEMS. 


49 


FIG.  56. 


/ 

{ 

y 

7 

* 

9 

^ 

'j_ 

'i 

)  d  cbaBabad 

FIG.  57. 

straight-edge  to  the  directrix  E  N, 
and  apply  to  it  a  square  LEG. 
Fasten  to  the  end  G  one  end  of  a 
thread  or  cord  equal  in  length  to  the 
edge  E  (7,  and  attach  the  other  end 
to  the  focus  F;  slide  the  square  along 
the  straight-edge,  holding  the  cord 
taut  against  the  edge  of  the  square 
by  a  pencil  D,  by  which  the  curve  is 
described. 

3d  Method:  When  the  height  and 
the  base  are  given  (Fig.  57).  —  Let 
A  B  be  the  given  axis,  and  C  D  a 
double  ordinate  or  base;  to  describe 
a  parabola  of  which  the  vertex  is  at 
A.  Through  A  draw  E  F  parallel  to 
C  D,  and  through  C  and  D  draw  C  E 
and  D  F  parallel  to  the  axis.  Divide 
B  C  arid  B  D  into  any  number  of 
equal  parts,  say  five,  at  a,  6,  etc.,  and 
divide  C  E  and  D  F  into  the  same 
number  of  parts.  Through  the 
points  a,  b,  c,  d  in  the  base  CD  on 
each  side  of  the  axis  draw  perpen- 
diculars, and  through  a,  b,  c,  d  in  C  E 
and  D  F  draw  lines  to  the  vertex  A , 
cutting  the  perpendiculars  at  e,  /,  g,  h. 
These  are  points  in  the  parabola,  and 
the  curve  CAD  may  be  traced  as 
shown,  passing  through  them. 
47.  The  Hyperbola  (Fig.  58).  —  A  hyperbola  is  a.  plane  curve,  such 
that  the  difference  of  the  distances  from  any  point  of  it  to  two  fixed  points 

is  equal  to  a  given  distance.  The 
fixed  points  are  called  the  foci. 

To  construct  a  hyperbola.  — 
Let  F/  and  F  be  the  foci,  and  Fe  F 
the  distance  Between  them.  Take  a 
ruler  longer  than  the  distance  F1  F, 
and  fasten  one  of  its  extremities  vj 
the  focus  F' .  At  the  other  extrem 
ity,  H,  attach  a  thread  of  such  a 
length  that  the  length  of  the  ruler 
shall  exceed  the  length  of  the  thread 
by  a  given  distance  A  B.  Attach 
the  other  extremity  of  the  thread  at 
the  focus  F. 

Press  a  pencil,  P,  against  the  ruler, 
and  keep  the  thread  constantly  tense, 
while  the  ruler  is  turned  around  F'  as 
a  centre.  The  point  of  the  pencil 
will  describe  one  branch  of  the  curve. 
2d  Method:  By  points  (Fig.  59).  — 
From  the  focus  F'  lay  off  a  distance 
F'  N  equal  to  the  transverse  axis,  or 
distance  between  the  two  branches  of 
the  curve,  and  take  any  other  dis- 
tance, as  F'  II,  greater  than  F'  N. 

With  F'  as  a  centre  and  F'  H  as  a 
radius  describe  the  arc  of  a  circle. 

hen  with  F  as  a  centre  and  N  H  as  a  radius  describe  an  arc  intersecting 
he  arc  before  described  at  p  and  q.     These  will  be  points  of  the  hyper- 
oia,  for  F'  q  —  F  q  is  equal  to  the  transverse  axis  A  B. 
If,  with  F  as  a  centre  and  F'  H  as  a  radius,  an  arc  be  described,  and  a 
second  arc  be  described  with  F'  as  a  centre  and  N  H  as  a  radius,  two  points 
in  the  other  branch  of  the  curve  will  be  determined.     Hence,  by  changing 
the  centres,  each  pair  of  radii  will  determine  two  points  in  each  branch. 
The  Equilateral  Hyperbola.  —  The  transverse  axis  of  a  hyperbola  is 


FIG.  58. 


\P/ 


FIG.  59. 


50 


GEOMETRICAL   PROBLEMS. 


the  distance,  on  a  line  joining  the  foci,  between  the  two  branches  of  the 
curve.  The  conjugate  axis  is  a  line  perpendicular  to  the  transverse  axis, 
drawn  from  its  centre,  and  of  such  a  length  that  the  diagonal  of  the  rect- 
angle of  the  transverse  and  conjugate  axes  is  equal  to  the  distance  between 
the  foci.  The  diagonals  of  this  rectangle,  indefinitely  prolonged,  are  the 
asymptotes  of  the  hyperbola,  lines  which  the  curve  continually  approaches, 
but  touches  only  at  an  infinite  distance.  If  these  asymptotes  are  perpen- 
dicular to  each  other,  the  hyperbola  is  called  a  rectangular  or  equilateral 
hyperbola.  It  is  a  property  of  this  hyperbola  that  if  the  asymptotes  are 
taken  as  axes  of  a  rectangular  system  of  coordinates  (see  Analytical  Geom- 
etry), the  product  of  the  abscissa  and  ordinate  of  any  point  in  the  curve  is 
equal  to  the  product  of  the  abscissa  and  ordinate  of  any  other  point ;  or,  if 
p  is  the  ordinate  of  any  point  and  v  its  abscissa,  and  p\,  and  vi  are  the 
ordinate  and  abscissa  of  any  other  point,  pv  =  p\v\\  or  pv  =  a  constant. 

48.  The  Cycloid  (Fig. 

60).  —If  a  circle  A  a  be  6  f 

rolled  along  a  straight 
line  A  6,  any  point  of  the 
circumference  as  A  will 
describe  a  curve,  which  is 
called  a  cycloid.  The 
circle  is  called  the  gene- 
rating circle,  and  A  the 
generating  point. 

To  draw  a  cycloid.  — 
Divide  the  circumference 
of  the  generating  circle 

into  an  even  number  of  equal  parts,  as  A  1, 12,  etc.,  and  set  off  these  dis- 
tances on  the  base.  Through  the  points  1,  2,  3,  etc.,  on  the  circle 
draw  horizontal  lines,  and  on  them 
set  off  distances  la  =  A 1 ,  2b  =  A  2,  3c  = 
A3,  etc.  The  points  A ,  a,  ft,  c,  etc., 
will  be  points  in  the  cycloid,  through 
which  draw  the  curve. 

49.  The  Epicycloid  (Fig.  61)  is 
generated  by  a  point  D  in  one  circle 
D  C  rolling  upon  the  circumference  of 
another  circle  A  C  B,  instead  of  on  a 
flat  surface  or  line;  the  former  being 
the  generating  circle,  and  the  latter 
the  fundamental  circle.   The  generat- 
ing circle  is  shown  in  four  positions, 
in    which    the   generating    point   is 
successively  marked  D,  D',  D",  D'". 
A  D'"  B  is  the  epicycloid. 


FIG.  61. 


50.  The  Hypocycloid  (Fig.  62) 
is  generated  by  a  point  in  the  gener- 
ating circle  rolling  on  the  inside  of 
the  fundamental  circle. 

When    the    generating    circle  = 
Tadius  of  the  other  circle,  the  hypo- 
cycloid  becomes  a  straight  line. 


51.  The  Tractrix  or  Schiele's 
anti-friction  curve  (Fig.  63).  —  R 
is  the  radius  of  the  shaft,  C,  1,  2,  etc., 
ihe  axis.  From  O  set  off  on  R  a 
rmall  distance,  oa;  with  radius  A  and 
centre  a  cut  the  axis  at  1,  join  a  1, 
and  set  off  a  like  small  distance  a  b; 
from  b  with  radius  R  cut  axis  at  2, 
join  b  2,  and  so  on,  thus  finding 
points  o,  a,  b,  c,  d,  etc.,  through  which 
the  curve  is  to  be  drawn. 


GEOMETRICAL   PROBLEMS. 


51 


52.  The  Spiral.  —  The  spiral  is  a  curve  described  by  a  point  which 
moves  along  a  straight  line  according  to  any  given  law,  the  line  at  the  same 
time  having  a  uniform  angular  motion.  The  line  is  called  the  radius  vector. 

If  the  radius  vector  increases  directly 
as  the  measuring  angle,  the  spires, 
or  parts  described  in  each  revolution, 
thus  gradually  increasing  their  dis- 
tance from  each  other,  the  curve  is 
known  as  the  spiral  of  Archimedes 


FIG.  64. 


his  curve  is  commonly  used  for 
cams.  To  describe  it  draw  the 
radius  vector  in  several  different 
directions  around  the  centre,  with 
equal  angles  between  them;  set  off 


the  distances  1,  2,  3,  4,  etc.,  corresponding  to  the  scale  upon  which  the 

curve  is  drawn,  as  shown  in  Fig.  64. 

In  the  common  spiral  (Fig.  64)  the 
pitch  is  uniform;  that  is,  the  spires 
are  equidistant.  Such  a  spiral  is 
made  by  rolling  up  a  belt  of  uniform 
thickness. 

To  construct  a  spiral  with  four 
centres  (Fig.  65).— Given  the 
pitch  of  the  spiral,  construct  a  square 
about  the  centre,  with  the  sum  of 
the  four  sides  equal  to  the  pitch. 
Prolong  the  sides  in  one  direction  as 
shown;  the  corners  are  the  centres  for 
each  arc  of  the  external^  angles, 
forming  a  quadrant  of  a  spire. 


FIG.  65. 


53.  To  find  the  diameter  of  a  circle  into  which  a  certain  number  of 
rings  will  fit  on  its  inside  (Fig.  66).  —  For  instance,  what  is  the  diameter 
of  a  circle  into  which  twelve  i/2-inch  rings  will  fit,  as  per  sketch?  Assume 
that  we  have  found  the  diameter  of  the  required  circle,  and  have  drawn 

the  rings  inside  of  it.  Join  the 
centres  of  the  rings  by  straight  lines, 
as  shown:  we  then  obtain  a  regular 
polygon  with  12  sides,  each  side 
being  equal  to  the  diameter  of  a 

fiven  ring.  We  have  now  to  find 
he  diameter  of  a  circle  circum- 
scribed about  this  polygon,  and  add 
the  diameter  of  one  ring  to  it;  the 
sum  will  be  the  diameter  of  the  circle 
into  which  the  rings  will  fit. 
Through  the  centres  A  and  D  of  two 
adjacent  rings  draw  the  radii  C  A 

R\(  \^  }  __       _,'    /  }/  and  CD;  since  the  polygon  has  twelve 

X-NC^^JT   S  sides  the  angle  A  C  D  =  30°  and 

N^^===^K^  AC  B  =  15°.     One  half  of  the  side 

^^^S^^^  A  D  is  equal  to  A  B.     We  now  give 

§7  the  following  proportion:  The  sine 

FIG.  66.  of  the  angle  A  C  B  is  to  A  B  as  1  is  to 

the  required  radius.     From  this  we 


. 

_  t  the  following  rule:  Divide  A  B  by  the  sine  of  the  angle  A  C  B\  the 
quotient  will  be  the  radius  of  the  circumscribed  circle;  add  to  the  corre- 
sponding diameter  the  diameter  of  one  ring;  the  sum  will  be  the  required 
diameter  F  G. 

54.  To  describe  an  arc  of  a  circle  which  is  too  large  to  be  drawn 
by  a  beam  compass,  by  means  of  points  in  the  arc,  radius  being  given. 
— Suppose  the  radius  is  20  feet  and  it  is  desired  to  obtain  five  points  in  an 
arc  whose  half  chord  is  4  feet.  Draw  a  line  equal  to  the  half  chord,  full 


uAvuiais  ui>  yuiiua  i,  2,  o,  uuu  •*  icci  iiuui   me  mat  pcipciiuicuiai.      ciuu 

Talues  of  y  in  the  formula  of  the  circle,  x*  *  j/a  »  R\  by  substituting  for 


52 


GEOMETRICAL  PROBLEMS. 


x  the  values  0,  1,  2,  3,  and  4,  etc.,  and  for  R2  the  square  of  the  radius,  or 
400.     The  values  will  be  y  =  ^R2  ~  x2  =  V400,  ^399,  V396,  V391, 

V384;  =  20,         19.975,     19.90,       19.774,       19.596. 
Subtract  the  smallest, 

or   19.596,   leaving  0.404,   0.379,        0.304,       0.178,         0  feet. 

Lay  off  these  distances  on  the  five  perpendiculars,  as  ordinates  from  the 
half  chord,  and  the  positions  of  five  points  on  the  arc  will  be  found. 
Through  these  the  curve  may  be 
drawn.  (See  also  Problem  14.) 

55.  The  Catenary  is  the  curve 
assumed  by  a  perfectly  flexible  cord 
when  its  ends  are  fastened  at  two 
points,  the  weight  of  a  unit  length 
being  constant. 

The  equation  of  the  catenary  is 

/  x       _?\ 
y=^(ea-}-e   a),  in  which  e  is  the 

base  of  the  Napierian  system  of  log- 
arithms. 

To  plot  the  catenary.  —  Let  o 
(Fig.  67)  be  the  origin  of  coordinates. 
Assigning  to  a  any  value  as  3,  the 
equation  becomes 


( 


To   find  the  lowest  point  of   the 
curve. 

Puts  =  0;   /.  y  =  - 


Then  put        x  =  \\  .'.  f—gl 

Put  z  =  2;  .'.  1/=|( 


H^  (1.396  +0.717)  =3.17. 
)  =  ?  (1.948  +0.513)  =3.69. 


Put  x  =  3,  4,  5,  etc.,  etc.,  and  find  the  corresponding  values  of  y.     For 
each  value  of  y  we  obtain  two  symmetrical  points,  as  for  example  p  and,  pr. 
In  this  way,  by  making  a  successively  equal  to  2,  3,  4,  5,  6,  7,  and  8,  the 
curves  of  Fig.  67  were  plotted. 

In  each  case  the  distance  from  the  origin  to 
the  lowest  point  of  the  curve  is  equal  to  a;  for 
putting  x  =  o,  the  general  equation  reduces  to 

For  values  of  a  —  6,  7,  and  8  the  catenary 
closely  approaches  the  parabola.  For  deriva- 
tion of  the  equation  of  the  catenary  see  Bow- 
ser's Analytic  Mechanics. 

56.  The  Involute  is  a  name  given  to  the 
curve  which  is  formed  by  the  end  of  a  string 
which  is  unwound  from  a  cylinder  and  kept 
taut;  consequently  the  string  as  it  is  unwound 
will  always  lie  in  the  direction  of  a  tangent 
to  the  cylinder.  To  describe  the  involute  of 
any  given  circle,  Fig.  68,  take  any  point  A  on 
its  circumference,  draw  a  diameter  A  B,  and 
from  B  draw  B  b  perpendicular  to  A  B.  Make 
B  b  equal  in  length  to  half  the  circumference 
of  the  circle.  Divide  B  b  nnd  the  semi-circum- 
ference into  the  same  number  of  equal  parts, 
say  six.  From  each  point  of  division  1,  2, 
3,  etc.,  on  the  circumference  draw  lines  to  the  centre  C  of  the  circle. 
Then  draw  lai  perpendicular  to  (71;  2  a^  perpendicular  to  (72;  and 
80  on.  Make  \a\  equal  to  b  6t;  2  dz  equal  to  b  62;  3  «a  equal  to  b  b&;  and 
so  on.  Join  the  points  A,  alt  ctz,  03,  etc.,  by  a  curve;  this  curve  will  be 
t&e  required  involute. 


FIG.  68. 


GEOMETRICAL   PROPOSITIONS.  53 


57.   Method  of  plotting  angles  without  using  a  protractor.  —  The 

radius  of  a  circle  whose  circumference  is  360  is  57.3  (more  accurately 
57.296).  Striking  a  semicircle  with  a  radius  57.3  by  any  scale,  spacers 
set  to  10  by  the  same  scale  will  divide  the  arc  into  18  spaces  of  10°  each 
and  intermediates  can  bo  measured  indirectly  at  the  rate  of  1  by  scale  for 
each  1°,  or  interpolated  by  eye  according  to  the  degree  of  accuracy  required 
The  following  table  shows  the  chords  to  the  above-mentioned  radius,  for 
every  10  degrees  from  0°  up  to  110°.  By  means  of  one  of  these  a  10° 
point  is  fixed  upon  the  paper  next  less  than  the  required  angle,  and  the 
remainder  is  laid  oft  at  the  rate  of  1  by  scale  for  each  degree. 

Angle.                Chord.  Angle.  Chord.  Angle.  Chord. 

1° 0.999      40° 39.192        80° 73658 

10° 9.988      50° 48.429  90°..,  81029 

20° 19.899      60° 57.296      100° '.  87>82 

30° 29.658      70° 65.727      110° 93.869 


GEOMETRICAL  PROPOSITIONS. 

In  a  right-angled  triangle  the  square  on  the  hypothenuse  is  equal  to  the 
sum  of  the  squares  on  the  other  two  sides. 

If  a  triangle  is  equilateral,  it  is  equiangular,  and  vice  versa. 

If  a  straight  line  from  the  vertex  of  an  isosceles  triangle  bisects  the  base, 
It  bisects  the  vertical  angle  and  is  perpendicular  to  the  base. 

If  one  side  of  a  triangle  is  produced,  the  exterior  angle  is  equal  to  the 
sum  of  the  two  interior  and  opposite  angles. 

If  two  triangles  are  mutually  equiangular,  they  are  similar  and  their 
corresponding  sides  are  proportional. 

If  the  sides  of  a  polygon  are  produced  in  the  same  order,  the  sum  of  the 
exterior  angles  equals  four  right  angles.  (Not  true  if  the  polygon  has 
re-entering  angles.) 

In  a  quadrilateral,  the  sum  of  the  interior  angles  equals  four  right 
angles. 

In  a  parallelogram,  the  opposite  sides  are  equal;  the  opposite  angles  are 
equal;  it  is  bisected  by  its  diagonal,  and  its  diagonals  bisect  each  other. 

If  three  points  are  not  in  the  same  straight  line,  a  circle  may  be  passed 
through  them. 

If  two  arcs  are  intercepted  on  the  same  circle,  they  are  proportional  to 
the  corresponding  angles  at  the  centre. 

If  two  arcs  are  similar,  they  are  proportional  to  their  radii. 

The  areas  of  two  circles  are  proportional  to  the  squares  of  their  radii. 

If  a  radius  is  perpendicular  to  a  chord,  it  bisects  the  chord  and  it  bisects 
the  arc  subtended  by  the  chord. 

A  straight  line  tangent  to  a  circle  meets  it  in  only  one  point,  and  it  13 
perpendicular  to  the  radius  drawn  to  that  point. 

If  from  a  point  without  a  circle  tangents  are  drawn  to  touch  the  circle, 
there  are  but  two;  they  are  equal,  and  they  make  equal  angles  with  the 
chord  joining  the  tangent  points. 

If  two  lines  are  parallel  chords  or  a  tangent  ,and  parallel  chord,  they 
intercept  equal  arcs  of  a  circle. 

If  an  angle  at  the  circumference  of  a  circle,  between  two  chords,  is  sub- 
tended by  the  same  arc  as  an  angle  at  the  centre,  between  two  radii,  tho 
angle  at  the  circumference  is  equal  to  half  the  angle  at  the  centre. 

If  a  triangle  is  inscribed  in  a  semicircle,  it  is  right-angled. 

If  two  chords  intersect  each  other  in  a  circle,  the  rectangle  of  the  seg- 
ments of  the  one  equals  the  rectangle  of  the  segments  of  the  other. 

And  if  one  chord  is  a  diameter  and  the  other  perpendicular  to  it,  the 
rectangle  of  the  segments  of  the  diameter  is  equal  to  the  square  on 
half  the  other  chord,  and  the  half  chord  is  a  mean  proportional  between 
the  segments  of  the  diameter. 

If  an  angle  is  formed  by  a  tangent  and  chord,  it  is  measured  by  one  half 
of  the  arc  intercepted  by  the  chord;  that  is,  it  is  equal  to  half  the  angle  at 
the  centre  subtended  by  the  chord. 


54  MENSURATION  —  PLANE   SURFACES. 


a  Railway  Curve.  —  This  last  proposition  is  useful  in  staking 
out  railway  curves.  A  curve  is  designated  as  one  of  so  many  degrees,  and 
the  degree  is  the  angle  at  the  centre  subtended  by  a  chord  of  100  ft.  To 
lay  out  a  curve  of  n  degrees  the  transit  is  set  at  its  beginning  or  "  point  of 
curve,"  pointed  in  the  direction  of  the  tangent,  and  turned  through  i/2?i 
degrees;  a  point  100  ft.  distant  in  the  line  of  sight  will  be  a  point  in  the 
curve.  The  transit  is  then  swung  1/2  n  degrees  further  and  a  100  ft.  chord 
is  measured  from  the  point  already  found  t9  a  point  in  the  new  line  of 
sight,  which  is  a  second  point  or  "  station  "  in  the  curve. 

The  radius  of  a  1°  curve  is  5729.65  ft.,  and  the  radius  of  a  curve  of  any 
degree  is  5729.65  ft.  divided  by  the  number  of  degrees. 

Some  authors  use  the  angle  subtended  by  an  arc  (instead  of  chord)  of 
100  ft.  in  defining  the  degree  of  a  curve.  For  a  statement  of  the  relative 
advantages  of  the  two  definitions,  see  Eng.  News,  Feb.  16,  1911. 

MENSURATION. 

PLANE  SURFACES. 

Quadrilateral*  —  A  four-sided  figure. 

Parallelogram.  —  A  quadrilateral  with  opposite  sides  parallel. 

Varieties.  —  Square:  four  sides  equal,  all  angles  right  angles.  Rect- 
angle: opposite  sides  equal,  all  angles  right  angles.  Rhombus:  four  sides 
equal,  opposite  angles  equal,  angles  not  right  angles.  Rhomboid:  opposite 
sides  equal,  opposite  angles  equal,  angles  not  right  angles. 

Trapezium.  —  A  quadrilateral  with  unequal  sides. 

Trapezoid.  —  A  quadrilateral  with  only  one  pair  of  opposite  sides 
parallel.  _ 

Diagonal  of  a  square  =  ^2  X  side2  =  1.4142  X  side.  _ 

Diag.  of  a  rectangle  =  v  sum  of  squares  of  two  adjacent  sides. 

Area  of  any  parallelogram  =  base  X  altitude. 

Area  of  rhombus  or  rhomboid  =  product  of  two  adjacent  sides  X  sine 
of  angle  included  between  them. 

Area  of  a  trapezoid  =  product  of  half  the  sum  of  the  two  parallel  sidea 
by  the  perpendicular  distance  between  them. 

To  find  the  area  of  any  quadrilateral  figure.  —  Divide  the  quad- 
rilateral into  two  triangles;  the  sum  of  the  areas  of  the  triangles  is  the 
area. 

Or,  multiply  half  the  product  of  the  two  diagonals  by  the  sine  of  the 
angle  at  their  intersection. 

To  find  the  area  of  a  quadrilateral  which  may  be  inscribed  in  a 
circle.  —  From  half  the  sum  of  the  four  sides  subtract  each  side  severally; 
multiply  the  four  remainders  together;  the  square  root  of  the  product  is 
the  area. 

Triangle.  —  A  three-sided  plane  figure. 

Varieties.  —  Right-angled,  having  one  right  angle;  obtuse-angled,  hav- 
ing one  obtuse  angle;  isosceles,  having  two  equal  angles  and  two  equal 
sides;  equilateral,  having  three  equal  sides  and  equal  angles. 

The  sum  of  the  three  angles  of  every  triangle  =  180°. 

The  sum  of  the  two  acute  angles  of  a  right-angled  triangle  =  90°. 

Hypothenuse  of  a  right-angled  triangle,  the  side  opposite  the  right 
angle,  =  Vsum  of  the  squares  of  the  other  two  sides.  If  a  and  6  are  the 
two  sides  and  c  the  hypothenuse,  c2=a2  +  &2;  a  =  Vc2-&2=V(c+&)(/-&). 

If  the  two  sides  are  equal,  side  =  hyp  -9-  1.4142;  or  hyp  X.7071. 

To  find  the  area  of  a  triangle  : 

RULE  1.   Multiply  the  base  by  half  the  altitude. 

RULE  2.  Multiply  half  the  product  of  two  sides  by  the  sine  of  the 
included  angle. 

RULE  3.  From  half  the  sum  of  the  three  sides  subtract  each  side 
severally;  multiply  together  the  half  sum  and  the  three  remainders,  and 
extract  the  square  root  of  the  product. 

The  area  of  an  equilateral  triangle  is  equal  to  one  fourth  _the  square  of 

one  of  its  sides  multiplied  by  the  square  root  of  3,  =»  a  .     ,  a  being  tht 
tide;  or  a8  X  0,433013, 


MENSURATION. 


55 


Area  of  a  triangle  given,  to  find  base:  Base  =  twice  area  •*•  perpendicular 
height. 

Area  of  a  triangle  given,  to  find  height:  Height  =  twice  area  -s-  base. 

Two  sides  and  base  given,  to  find  perpendicular  height  (in  a  triangle  in 
which  both  of  the  angles  at  the  base  are  acute). 

RULE.  —  As  the  base  is  to  the  sum  of  the  sides,  so  is  the  difference  of  the 
sides  to  the  difference  of  the  divisions  of  the  base  made  bv  drawing  the 
perpendicular.  Half  this  difference  being  added  to  or  subtracted  from 
half  the  base  will  give  the  two  divisions  there9f.  As  each  side  and  its 
opposite  division  of  the  base  constitutes  a  right-angled  triangle.,  the 
perpendicular  is  ascertained  by  the  rule:  Perpendicular  =  Vhyp2  —  base2* 

Areas  of  similar  figures  are  to  each  other  as  the  squares  of  their 
respective  linear  dimensions.  If  the  area  of  an  equilateral  triangle  of 
side  =  1  is  0.433013  and  its  height  0.86603,  what  is  the  area  of  a  similai 
triangle  whose  height  =  1?  0.866032  :  I2  ::  0.433013  :  0.57735,  Ans. 

Polygon.  —  A  plane  figure  having  three  or  more  sides.  Regular  or 
irregular,  according  as  the  sides  or  angles  are  equal  or  unequal.  Polygons 
are  named  from  the  number  of  their  sides  and  angles. 

To  find  the  area  of  an  irregular  polygon.  —  Draw  diagonals  dividing 
the  polygon  into  triangles,  and  find  the  sum  of  the  areas  of  these  triangles. 

To  find  the  area  of  a  regular  polygon: 

RULE.  —  Multiply  the  length  of  a  side  by  the  perpendicular  distance  to 
the  centre;  multiply  the  product  by  the  number  of  sides,  and  divide  it  by 
2.  Or,  multiply  half  the  perimeter  by  the  perpendicular  let  fall  from  the 
centre  on  one  of  the  sides. 

The  perpendicular  from  the  centre  is  equal  to  half  of  one  of  the  sides  of 
the  polygon  multiplied  by  the  cotangent  of  the  angle  subtended  by  the 
half  side. 

The  angle  at  the  centre  =  360°  divided  by  the  number  of  sides. 


Table  of  Regular  Polygons^ 


«H 

Radius  of  Cir- 

II 

cumscribed 

12  • 

c3  o 

t 

d 

ft 

Circle. 

^Q1"1 

^  d 

2 

1 

1 

iH 

£ 

~ 

K3 
t 

1.1 

2  ® 

C-TJ 

si 

i;' 

1 

"d 

6 

S  $ 

m 

1- 

w 

*o 

02 

J 

i—  i  II 

i"1 

°« 

S^^j 

"S 

0)^9 

^  a 

o 

g 

OQ 

o£ 

II 

11 

"S'S  2 

o 

^  S 

& 

i 

1 

1 

•   £ 

1 

J'3 

1 

r 

3 

Triangle 

0.4330 

0.5773 

2.000 

0.5773 

0.2887 

1.732 

120° 

60° 

4 

Square 

1.0000 

1.0000 

.414 

0.7071 

0.5000 

1.4142 

90 

90 

5 

Pentagon 

1  .  7205 

0.7265 

.236 

0.8506 

0.6882 

1  .  1  756 

72 

108 

6 

Hexagon 

2.5981 

0.8660 

.155 

1  .  0000 

0.866 

1  .  0000 

60 

120 

7 

Heptagon 

3.6339 

0.7572 

.11 

1  .  1  524 

1  .  0383 

0.8677 

51  26' 

1284-7 

8 

Octagon 

4.8284 

0.8284 

.082 

1  .  3066 

.2071 

0.7653 

45 

135 

9 

Nonagon 

6.1818 

0.7688 

.064 

1  4619 

.3737 

0.684 

40 

140 

10 
11 

Decagon 
Undecagon 

7.6942 
9.3656 

0.8123 
0.7744 

.051 
.042 

1.618 
1  .  7747 

.5388 
.7028 

0.618 
0.5634 

36 
3243' 

144 
1473-11 

12 

Dodecagon 

11.1962 

0.8038 

.035 

1.9319 

.866 

0.5176 

30 

150 

*  Short  diameter,  even  number  of  sides,  =  diam.  of  inscribed  circle: 
short  diam.,  odd  number  of  sides,  =  rad.  of  inscribed  circle  +  rad.  ol 
circumscribed  circle. 


56 


AREA   OF   IRREGULAR  FIGURES. 


To  find  the  area  of  a  regular  polygon,  when  the  length  of  a  side 
only  is  given: 

RULE.— Multiply  the  square  of  the  side  by  the  figure  for  "area,  side  — 
1,"  opposite  to  the  name  of  the  polygon  in  the  table. 

Length  of  a  side  of  a  regular  polygon  inscribed  in  a  circle  =  diam. 
X  sin  (180°  •*-  no.  of  sides). 


No.  of  sides  sin  (180° /n)          No.  sin  (180° /n) 


0.86603 
.70711 
.58778 
.50000 
.43388 
.38268 


9  0.34202 

10  .30902 

11  .28173 

12  .25882 

13  .23931 

14  .22252 


No.  sin  (180°/n) 

15  0.20791 

16  .19509 

17  .18375 

18  .17365 

19  .16458 

20  .15643 


To  find  the  area  of  an  irregular 
i^gure  (Fig.  69).  —  Draw  ordinates 
f, cross  its  breadth  at  equal  distances 
apart,  the  first  and  the  last  ordinate 
each  being  one  half  space  from  the 
ends  of  the  figure.  Find  the  average 
breadth  by  adding  together  the 
lengths  of  these  lines  included  be- 
tween the  boundaries  of  the  figure, 
and  divide  by  the  number  of  the  lines 
added;  multiply  this  mean  breadth 
by  the  length.  The  greater  the  num- 
ber of  lines  the  nearer  the  approxi- 
mation. 


\l*  3  4  $ 


£ 


FIG.  69. 


In  a  figure  of  very  irregular  outline,  as  an  indicator-diagram  from  a 
high-speed  steam-engine,  mean  lines  may  be  substituted  for  the  actual 
lines  of  the  figure,  being  so  traced  as  to  intersect  the  undulations,  so  that 
the  total  area  of  the  spaces  cut  off  may  be  compensated  by  that  of  the 
extra  spaces  inclosed. 

2d  Method:  THE  TRAPEZOIDAL  RULE.  —  Divide  the  figure  into  any 
sufficient  number  of  equal  parts;  add  half  the  sum  of  the  two  end  ordinates 
to  the  sum  of  all  the  other  ordinates;  divide  by  the  number  of  spaces 
(that  is,  one  less  than  the  number  of  ordinates)  to  obtain  the  mean 
ordinate,  and  multiply  this  by  the  length  to  obtain  the  area. 

3d  Method:  SIMPSON'S  RULE.  —  Divide  the  length  of  the  figure  into  any 
even  number  of  equal  parts,  at  the  common  distance  D  apart,  and  draw 
ordinates  through  the  points  of  division  to  touch  the  boundary  lines 
Add  together  the  first  and  last  ordinates  and  call  the  sum  A ;  add  together 
the  even  ordinates  and  call  the  sum  J5;  add  together  the  odd  ordinates, 
except  the  first  and  last,  and  call  the  sum  C.  Then, 


area  of  the  figure  = 


A+4B+2C 


XD. 


4/fe  Method:  DURAND'S  RULE.  —  Add  together  */io  the  sum  of  the  first 
and  last  ordinates,  1  Vio  the  sum  of  the  second  and  the  next  to  the  last 
(or  the  penultimates),  and  the  sum  of  all  the  intermediate  ordinates. 
Multiply  the  sum  thus  gained  by  the  common  distance  between  the  ordi- 
nates to  obtain  the  area,  or  divide  this  sum  by  the  number  of  spaces  to 
f  btain  the  mean  ordinate. 

Prof.  Durand  describes  the  method  of  obtaining  his  rule  in  Engineering 
News,  Jan.  18,  1894.  He  claims  that  it  is  more  accurate  than  Simpson's 
rule,  and  practically  as  simple  as  the  trapezoidal  rule.  He  thus  describes 
its  application  for  approximate  integration  of  differential  equations.  Any 
definite  integral  may  be  represented  graphically  by  an  area.  Thus,  let 

Q  =  fu  dx 


be  an  integral  in  which  u  is  some  function  of  x,  either  known  or  admitting 
of  computation  or  measurement.  Any  curve  plotted  with  x  as  abscissa 
and  u  as  ordinate  will  then  represent  the  variation  of  u  with  x,  and  tht 


MENSURATION. 


57 


area  between  such  curve  and  the  axis  X  will  represent  the  integral  in 
question,  no  matter  how  simple  or  complex  may  be  the  real  nature  of  the 
function  u. 

Substituting  in  the  rule  as  above  given  the  word  "  volume"  for  "  area" 
and  the  W9rd  "section"  for  "  ordinate,"  it  becomes  applicable  to  the 
determination  of  volumes  from  equidistant  sections  as  well  as  of  areas 
from  equidistant  ordinates. 

Having  approximately  obtained  an  area  by  the  trapezoidal  rule,  the 
area  by  Durand's  rule  may  be  found  by  adding  algebraically  to  the  sum  of 
the  ordinates  used  in  the  trapezoidal  rule  (that  is,  half  the  sum  of  the  end 
ordinates  -f  sum  of  the  other  ordinates)  1/10  of  (sum  of  penultimates 
—  sum  of  first  and  last)  and  multiplying  by  the  common  distance  between 
the  ordinates. 

5ih  Method.  —  Draw  the  figure  on  cross-section  paper.  Count  the 
number  of  squares  that  are  entirely  included  within  the  boundary;  then 
estimate  the  fractional  parts  of.  squares  that  are  cut  by  the  boundary,  add 
together  these  fractions,  and  add  the  sum  to  the  number  of  whole  squares. 
The  result  is  the  area  in  units  of  the  dimensions  of  the  squares.  The  finer 
the  ruling  of  the  cross-section  paper  the  more  accurate  the  result. 

6th  Method.  —  Use  a  planimeter. 

7th  Method.  —  With  a  chemical  balance,  sensitive  to  one  milligram, 
draw  the  figure  on  paper  of  uniform  thickness  and  cut  it  out  carefully; 
weigh  the  piece  cut  out,  and  compare  its  weight  with  the  weight  per 
square  inch  of  the  paper  as  tested  by  weighing  a  piece  of  rectangular  shape. 


THE  CIRCLE. 


Circumference  =  diameter  X  3.  1416,  nearly;  more  accurately,  3.14159265359. 

99  "^^^ 

Approximations,         =  3.143;  =  3.1415929. 


The  ratio  of  circum.  to  diam.  is  represented  by  the  symbol 
Area  =  0.7854   X  square  of  the  diameter. 


(called  Pi). 


Multiples  of  »r. 

1*  =  3.14159265359 
In  =  6.28318530718 
37r  =  9.42477796077 
4*  =  12.56637061436 
5x  =  15.70796326795 
6^  =  18.84955592154 
In  =  21.99114857513 
8;:  =  25.13274122872 
9*  =  28.27433388231 


7T/4 


Multiples  of|- 

=  0.7853982 
X  2  =  1.5707963 
X  3  =  2.356194r 
X  4  =  3.1415927 
X  5  =  3.9269908 
X  6  =  4.7123890 
X  7  =  5.4977871 
X  8  =  6.2831853 
X  9  =  7.0685835 


Ratio  of  diam.  to  circumference  =  reciprocal  of  «  =  0.3183099. 

1/7^=0.101321 
VK=  1.772453 
V7/7  =0.564189 
vV/4  =0.886226 
LogTr    =0.497 14987 
Log  ir/4_=  1.895090 
Log  vV  =0.248575 
Log  vV/4=  1.947545 


iprocal  of  »/4  =  1.27324. 

10/7r=     3.18310 

Multiples  of  I/TT. 

12/x=     3.81972 

I/TT  =  0.31831 

x/2  =      1.570796 

2/7T  =  0.63662 

7T/3  =      1.047197 

3/?r  =  0.95493 

7T/6  =     0.523599 

4/7r=  1.27324 

7T/12  =     0.261799 

5/7T  =  1.59155 

ir/64  =      0.049087 

6/7r=  1.90986 

Tr/360  =      0.0087266 

7/7r=  2.22817 

360/7r=  114.5915 

8/7T  =  2.54648 

*•*  =      9.86960 

9/7T  =  2.86479 

1-^-4*-=  0.0795775 

Diam.  in  ins.  =  13.5405  Varea  in  sq.  ft. 

Area  in  sq.  ft.  =  (diam  in  inches)2  X  .0054542. 

D  =  diameter,     R  =  radius,     C  =  circumference, 


;  area. 


58 


THE   CIRCLE. 


.  =      ;  =  .0795802  ;=        - 


R  =  •- 


0.31831(7; 


0.159155C; 


;  =  2  4/-;  =  1. 

V    * 


12838 


~  ;  =  0.564189 


Areas  of  circles  are  to  each  other  as  the  squares  of  their  diameters. 
To  find  the  length  of  an  arc  of  a  circle: 

RULE  1.  As  360  is  to  the  number  of  degrees  in  the  arc,  so  is  the  circum- 
ference of  the  circle  to  the  length  of  the  arc. 

RULE  2.  Multiply  the  diameter  of  the  circle  by  the  number  of  degrees 
in  the  arc,  and  this  product  by  0.0087266. 


Relations  of  Arc,  Chord,  Chord  of  Half  the  Arc,  etc. 

Let  R  =  radius,  D  =  diameter,  L  =  length  of  arc, 
C  =  chord  of  the  arc,  c  =  chord  of  half  the  arc, 
V  =  rise,  or  height  of  the  arc, 

9/>  V  1  0  F 


Length  of  the  arc  =  L 


-  (very  nearly),  = 


+  2c'  nearly» 


4F2X 


15CS+33FS 
Chord  of  the  arc  C,  =  2  >/c2  -  F2;  = 


..  nearly. 


-  (D  -  2F)2;  =  8c  -  3L 
=  2  \/(D  -  F)  X  F. 
Chord  of  half  the  arc,  c  =  i/2  v/<72+  4F2;  =  VD  x  F;  =  (3L  -f  C)  •*•  8. 
Diameter  of  the  circle,  D  =       ;=   V4  C24-  F^; 


Rise  of  the  arc,  F  =  ^  ;  =  1/2  (D  -  ' 

(or  if  F  is  greater  than  radius     1/2  (I>  +  ' 


-  <72)  ; 


Half  the  chord  of  the  arc  is  a  mean  proportional  between  the  rise  and 
the  diameter  minus  the  rise:  1/2  C  =  V'F  X  (  £  -  F). 

Length  of  the  Chord  subtending  an  angle  at  the  centre  =  twice  the 
sine  of  half  the  angle.  (See  Table  of  Sines.) 

Ordinates  to  Circular  Arcs.  —  C  =  chord,  F  =  height  of  the  arc,  or 
middle  ordinate,  x  =  abscissa,  or  distance  measured  on  the  chord  from  its 
central  point,  y  =  ordinate,  or  distance  from  the  arc  to  the  chord  at  the 
point  x,  V  =  R  -  ^R2  -  1/4C'2;  y  =  ^R2  -  x2  -  (R  -  F). 


Length  of  a  Circular  Arc.  —  Huyghens's  Approximation. 

Length  of  the  arc,  L  =  (8c  —  C)  •*•  3.  Professor  Williamson  shows 
that  when  the  arc  subtends  an  angle  of  30°,  the  radius  being  100,000  feet 
(nearly  19  miles),  the  error  by  this  formula  is  about  two  inches,  or  1/600000 
part  of  the  radius.  When  the  length  of  the  arc  is  equal  to  the  radius,  i.e., 
when  it  subtends  an  angle  of  57°.  3,  the  error  is  less  than  1/7680  part  of  the 
radius.  Therefore,  if  the  radius  is  100,000  feet,  the  error  is  less  than 
100000/7680  =  13  feet.  The  error  increases  rapidly  with  the  increase  of 
the  angle  subtended.  For  an  arc  of  120°  the  error  is  1  part  in  400;  for  an 
arc  of  180°  the  error  is  1.18%, 


MENSURATION. 


59 


In  the  measurement  of  an  arc  which  is  described  with  a  short  radius  the 
error  is  so  small  that  it  may  be  neglected.  Describing  an  arc  with  a  radius 
of  12  inches  subtending  an  angle  of  30°,  the  error  is  1/50000  of  an  inch. 

To  measure  an  arc  when  it  subtends  a  large  angle,  bisect  it  and  measure 
each  half  as  before — in  this  case  making  B  =•  length  of  the  chord  of  half  the 
arc,  and  b = length  of  the  chord  of  one  fourth  the  arc;  then  L  =  (166  -  25)  -*-  3. 

Formulas  for  a  Circular  Curve. 

J.  C.  Locke,  Eng.  News,  March  16,  1908. 
c 


u 


;-  =  ^2R  (R-  V(R  +&)(#_  6) 
=  2\Sm  (2R  —  m),  =  2R  sin  1/27, 
=  2 17  cos  1/2  7. 

e  =  R  exsec  1/27,  =»  R  tan  l/27  tan  1/47, 
=  T  tan  1/4  7. 


7i)sin7l  =  a  cot  1/2  7. 


2a 


2m  ' 


-  c)  (2R  -  c)),  =  2R  sin  1/47. 
Y  =  R  vers  i/27, 


JK  sin  1/2  /  tan  1/4  /,  =  1/2  c  tan  1/4  /. 


-£if 


+6)  (fi  - 


(sin  1/2  7)2,  =  R  vers  7, 


R  sin  7  tan  1/27,  =  &  tan  1/27,  =••  T  sin  7. 
=  #tani/27.  r      L^ 


I  =  ±L  x  57.295780°. 

I  —  c 


=  IR  X  0.01745329, 


Area  of  Segment  =  —  --  • 


2  sin  7 


X  57.295780°. 


1Z& 
2  * 


Relation  of  the  Circle  to  its  Equal,  Inscribed,  and  Circum- 
scribed Squares. 


Diameter  of  circle  X 

Circumference  of  circle  X 
Circumference  of  circle  X 
Diameter  of  circle  X 

Circumference  of  circle  X 
Area  of  circle  X  0.90031  -f- 
Area  of  circle  X 
Area  of  circle  X 
Side  of  square  X 
X 

"       X 
X 

Perimeter  of  square  X 
Square  inches  X 


0.88623  ) 
0.28209  J 
1.1284 
0.7071    ) 
0.22508}     = 
liameter) 


diameter 
1.2732 
0.63662 
1.4142 
4.4428 
1.1284 
3.5449 
0.88623 
1.2.732 


side  of  equal  square, 
perimeter  of  equal  square. 

side  of  inscribed  square. 

=  area  of  circumscribed  square. 
=  area  of  inscribed  square. 
=  diam.  of  circumscribed  circle. 
=  circum. 

=»  diam.  of  equal  circle. 
«=  circum.        ^         ^ 

=  circular  inches. 


GO  MENSURATION. 

Sectors  and  Segments.  —  To  find  the  area  of  a  sector  of  a  circle. 

RULE  1.   Multiply  the  arc  of  the  sector  by  half  its  radius. 

RULE  2.  As  360  is  to  the  number  of  degrees  in  the  arc,  so  is  the  area  of 
the  circle  to  the  area  of  the  sector. 

RULE  3.  Multiply  the  number  of  degrees  in  the  arc  by  the  square  of  the 
radius  and  by  0.008727. 

To  find  the  area  of  a  segment  of  a  circle:  Find  the  area  of  the  sector 
which  has  the  same  arc,  and  also  the  area  of  the  triangle  formed  by  the 
chord  of  the  segment  and  the  radii  of  the  sector. 

Then  take  the  sum  of  these  areas,  if  the  segment  is  greater  than  a  semi- 
circle, but  take  their  difference  if  it  is  less.  (See  Table  of  Segments.) 

Another  Method:  Ar^a  of  segment  =  V2.R2  (arc  —  sin  A),  in  which  A  is 
the  central  angle,  R  the  radius,  and  arc  the  length  of  arc  to  radius  1  . 

To  find  the  area  of  a  segment  of  a  circle  when  its  chord  and  height  only 
are  given.  First  find  radius,  as  follows: 


radius  -  1  [sq^e  °f™^  ChOrd  +  height  ]  . 

2.  Find  the  angle  subtended  by  the  arc,  as  follows:    half  chord  •*• 
radius  =  sine  of  half  the  angle.     Take  the  corresponding  angle  from  a 
table  of  sines,  and  double  it  to  get  the  angle  of  the  arc. 

3.  Find  area  of  the  sector  of  which  the  segment  is  a  part: 

area  of  sector  =  area  of  circle  X  degrees  of  arc  -*•  360. 

4.  Subtract  area  of  triangle  under  the  segment: 

Area  of  triangle  =  half  chord  X  (radius  —  height  of  segment).     . 

The  remainder  is  the  area  of  the  segment. 

When  the  chord,  arc,  and  diameter  are  given,  to  find  the  area.  From 
the  length  of  the  arc  subtract  the  length  of  the  chord.  Multiply  the 
remainder  by  the  radius  or  one-half  diameter;  to  the  product  add  the 
chord  multiplied  by  the  height,  and  divide  the  sum  by  2. 

Given  diameter,  d,'and  height  of  segment,  h. 


When  h  is  from  0  to  1/4  c?,  area      =  feVl.766(/fe  -  fe2; 

1/2  d,  area  =  h\/Q.Ol7d2  +  \.ldh  -  h2 


(approx.).     Greatest  error  0.23%,  when  h  =  i/4rf. 

To  find  the  chord:  From  the  diameter  subtract  the  height;  multiply 
the  remainder  by  four  times  the  height  and  extract  the  square  root. 

When  the  chords  of  the  arc  and  of  half  the  arc  and  the  rise  are  given: 
To  the  chord  of  the  arc  add  four  thirds  of  the  chord  of  half  the  arc;  mul- 
tiply the  sum  by  the  rise  and  the  product  by  0.40426  (approximate). 

Circular  Ring.  —  To  find  the  area  of  a  ring  included  between  the  cir- 
cumferences of  two  concentric  circles:  Take  the  difference  between  the.  areas 
of  the  two  circles;  or,  subtract  the  square  of  the  less  radius  from  the  square 
of  the  greater,  and  multiply  their  difference  by  3.14159. 

The  area  of  the  greater  circle  is  equal  to  nR*; 
and  the  area  of  the  smaller,  ~r2. 

Their  difference,  or  the  area  of  the  ring,  is  n(R*  -  r2). 
The  Ellipse.  —  Area  of  an  ellipse  =  product  of  its  semi-axes  X3.14159 

=  product  of  its  axes  X  0.785398. 

The  Ellipse.  —  Circumference  (approximate)  =  3.1416   y  -  -  —  ,   D 

and  d  being  the  two  axes. 

Trautwine  gives  the  following  as  more  accurate:  When  the  longer  axis 
D  is  not  more  than  five  times  the  length  of  the  shorter  axis,  dt 


Circumference  -  3.1416 


MENSURATION.  61 

"When  D  is  more  than  5d,  the  divisor  8.8  is  to  be  replaced  by  the  fallowings 

ForD/d  =  6     789        10     12    14      16      18       20     30      40        50 
Divisor    =  9   9.2   9.3   9.35  9.4  9.5  9.6  9.68   9.75   9.8   9.92   9.98       10 


in  which  A  =          -  —  Ingenieurs  Taschenbuch,  1896.    (a  and  6,  semi-axes.) 


Carl  G.  Earth  (Machinery,  Sept.,  1900)  gives  as  a  very  close  approxi- 
mation to  this  formula 


Area  of  a  segment,  of  an  ellipse  the  base  of  which  is  parallel  to  one  of 
the  axes  of  the  ellipse.  Divide  the  height  of  the  segment  by  the  axis  of 
which  it  is  part,  and  find  the  area  of  a  circular  segment,  in  a  table  9f  circu- 
lar segments,  of  which  the  height  is  equal  to  the  quotient;  multiply  the 
area  thus  found  by  the  product  of  the  two  axes  of  the  ellipse. 

Cycloid.  —  A  curve  generated  by  the  rolling  of  a  circle  on  a  plane. 

Length  of  a  cycloidal  curve  =  4  X  diameter  of  the  generating  circle. 
Length  of  the  base  =  circumference  of  the  generating  circle. 
Area  of  a  cycloid  =  3  X  area  of  generating  circle. 

Helix  (Screw).  —  A  line  generated  by  the  progressive  rotation  cf  a 
point  around  an  axis  and  equidistant  from  its  center. 

Length  of  a  helix.  —  To  the  square  of  the  circumference  described  by  the 
generating  point  add  the  square  of  the  distance  advanced  in  one  revolution, 
and  take  the  square  root  of  their  sum  multiplied  by  the  number  of  revolu- 
tions of  the  generating  point.  Or, 


«  length,  n  being  number  of  revolutions. 

Spirals.  —  Lines  generated  by  the  progressive  rotation  of  a  point 
around  a  fixed  axis,  with  a  constantly  increasing  distance  from  the  axis. 

A  plane  spiral  is  made  when  the  point  rotates  in  one  plane. 

A  conical  spiral  is  made  when  the  point  rotates  around  an  axis  at  a 
progressing  distance  from  its  center,  and  advancing  in  the  direction  of  the 
axis,  as  around  a  cone. 

Length  of  a  plane  spiral  line.  —  When  the  distance  between  the  coils  is 
uniform. 

RULE.  —  Add  together  the  greater  and  less  diameters;  divide  their  sum 
by  2;  multiply  the  quotient  by  3.1416,  and  again  by  the  number  of  revo- 
lutions. Or,  take  the  mean  of  the  length  of  the  greater  and  less  circum- 
ferences and  multiply  it  by  the  number  of  revolutions.  Or, 

length  =  im(R  +r),  R  and  r  being  the  outer  and  inner  radii.    To  find  n, 

let  t  =  thickness  of  coil  or  band,  s  =  space  between  the  coils,  n  =    .  . — •• 

i  ~r  s 

Length  of  a  conical  spiral  line.  —  Add  together  the  greater  and  less 
diameters;  divide  their  sum  by  2  and  multiply  the  quotient  by  3.1416. 
To  the  square  of  the  product  of  this  circumference  and  the  number  of 
revolutions  of  the  spiral  add  the  square  of  the  height  of  its  axis  and  take 
the  square  root  of  the  sum. 


Or,  length 


SOLID  BODIES. 

Surfaces  and  Volumes  of  Similar  Solids.  —  The  surfaces  of  two 
similar  solids  are  to  each  other  as  the  squares  of  their  linear  dimensions; 
the  volumes  are  as  the  cubes  of  their  linear  dimensions.  If  L  =  the  side 


62  MENSURATION. 

of  a  cube  or  other  solid,  and  /  the  side  of  a  similar  body  of  different  size, 
S,  s,  the  surfaces  and  V,  v,  the  volumes  respectively,  S  :  s  ::  L2 :  /*; 
V  :  v  ::  L3  :  J«. 

The  Prism.  —  To  find  the  surface  of  a  right  prism:  Multiply  the  perim- 
eter of  the  base  by  the  altitude  for  the  convex  surface.  To  this  add  the 
areas  of  the  two  ends  when  the  entire  surface  is  required. 

Volume  of  a  prism  =  area  of  its  base  X  its  altitude. 

The  pyramid.  —  Convex  surface  of  a  regular  pyramid  =  perimeter  of 
its  base  X  half  the  slant  height.  To  this  add  area  of  the  base  if  the  whole 
surface  is  required. 

Volume  of  a  pyramid  =  area  of  base  X  one  third  of  the  altitude. 

To  find  the  surface  of  a  frustum  of  a  regular  pyramid:  Multiply  half  the 
slant  height  by  the  sum  of  the  perimeters  of  the'  two  bases  for  the  convex 
surface.  To  this  add  the  areas  of  the  two  bases  when  the  entire  surface  is 
required . 

To  find  the  volume  of  a  frustum  of  a  pyramid:  Add  together  the  areas  of 
the  two  bases  and  a  mean  proportional  between  them,"  and  multiply  the 
sum  by  one  third  of  the  altitude.  (Mean  proportional  between  two 
numbers  =  square  root  of  their  product.) 

Wedge.  —  A  wedge  is  a  solid  bounded  by  five  planes,  viz.:  a  rectangular 
base,  two  trapezoids,  or  two  rectangles,  meeting  in  an  edge,  and  two 
triangular  ends.  The  altitude  is  the  perpendicular  drawn  from  any  point 
in  the  edge  to  the  plane  of  the  base. 

To  find  the  volume  of  a  wedge:  Add  the  length  of  the  edge  to  twice  the 
length  of  the  base,  and  multiply  the  sum  by  one  sixth  of  the  product  of 
the  height  of  the  wedge  and  the  breadth  of  the  base. 

Rectangular  prismoid.  —  A  rectangular  prisrnoid  is  a  solid  bounded 
by  six  planes,  of  which  the  two  bases  are  rectangles,  having  their  corre- 
sponding sides  parallel,  and  the  four  upright  sides  of  the  solid  are  trape- 
zoids. 

To  find  the  volume  of  a  rectangular  prismoid:  Add  together  the  areas  of 
the  two  bases  and  four  times  the  area  of  a  parallel  section  equally  distant 
from  the  bases,  and  multiply  the  sum  by  one  sixth  of  the  altitude. 

Cylinder.  —  Convex  surface  of  a  cylinder  =  perimeter  of  base  X 
altitude.  To  this  add  the  areas  of  the  two  ends  when  the  entire  surface  is 
required. 

Volume  of  a  cylinder  —  area  of  base  X  altitude. 

Cone.  —  Convex  surface  of  a  cone  =  circumference  of  base  X  half  the 
slant  height.  To  this  add  the  area  of  the  base  when  the  entire  surface  is 
required. 

Volume  of  a  cone  =  area  of  base  X  one  third  of  the  altitude. 

To  find  the  surface  of  a  frustum  of  a  cone:  Multiply  half  the  side  by  the 
sum  of  the  circumferences  of  the  two  bases  for  the  convex  surface;  to  this 
add  the  areas  of  the  two  bases  when  the  entire  surface  is  required. 

To  find  the  volume  of  a  frustu?n  of  a  cone:  Add  together  the  areas  of 
the  two  bases  and  a  mean  proDortional  between  them,  and  multiply 
the  sum  by  one  third  of  the  altitude.  Or,  Vol.  =  0.261Sa(624-  c2  +  be); 
a  =  altitude;  b  and  c,  diams.  of  the  two  bases. 

Sphere.  —  To  find  the  surface  of  a  sphere:  Multiply  the  diameter  by  the 
circumference  of  a  great  circle;  or,  multiply  the  square  of  the  diameter  by 
3.14159. 

Surface  of  sphere  —  4  x  area   of  its  great  circle. 
*'  *'        **        =i  convex  surface  of  its  circumscribing  cylinder. 

Surfaces  of  spheres  are  to  each  other  as  the  squares  of  their  diameters. 
To  find  the  volume  of  a  sphere:  Multiply  the  surface  by  one  third  of  the 
radius;  or,  multiply  the  cube  of  the  diameter  by  ;r/6;  that  is,  by  0.5236, 
Value  of  7T/6  to' 10  decimal  places  =  0.5235987756. 
The  volume  of  a  sphere  =  2/3  the  volume  of  its  circumscribing  cylinder. 
Volumes  of  spheres  are  to  each  other  as  the  cubes  of  their  diameters. 


MENSURATION.  63 


Spherical  triangle.  —  To  find  the  area  of  a  spherical  triangle:  Compute 
the  surface  of  the  quadrantal  triangle,  or  one  eighth  of  the  surface  of 
the  sphere.  From  the  sum  of  the  three  angles  subtract  two  right  angles; 
divide  the  remainder  by  90,  and  multiply  the  quotient  by  the  area  of  the 
quadrantal  triangle. 

Spherical  polygon.  —  To  find  the  area  of-a  spherical  polygon:  Compute 
the  surface  of  the  quadrantal  triangle.  From  the  sum  of  all  the  angles 
subtract  the  product  of  two  right  angles  by  the  number  of  sides  less  two; 
divide  the  remainder  by  90  and  multiply  the  quotient  by  the  area  of  the 
quadrantal  triangle. 

The  prismoid.  —  The  prismoid  is  a  solid  having  parallel  end  areas,  and 
may  be  composed  of  any  combination  of  prisms,  cylinders,  wedges,  pyra- 
mids, or  cones  or  frustums  of  the  same,  whose  bases  and  apices  lie  in  the 
end  areas. 

Inasmuch  as  cylinders  and  cones  are  but  special  forms  of  prisms  and 
pyramids,  and  warped,  surface  solids  may  be  divided  into  elementary 
forms  of  them,  and  since  frustums  may  also  be  subdivided  into  the  elemen- 
tary forms,  it  is  sufficient  to  say  that  all  prismoids  may  be  decomposed 
into  prisms,  wedges,  and  pyramids.  If  a  formula  can  be  found  which  is 
equally  applicable  to  all  of  these  forms,  then  it  will  apply  to  any  combi- 
nation of  them.  {Such  a  formula  is  called 


The  Prismoictal  Formula. 

Let  A  =   area  of  the  base  of  a  prism,,  wedge,  or  pyramid: 
Ai,  Azt  Am  =  the  two  end  and  the  middle  areas  of  a  prismoid,  or  of  any  ol 
its  elementary  solids;  h  =  altitude  of  the  prismoid  or  elementary  solid? 
V  =  its  volume; 


For  a  prism,  Ai,  Am  and  A*  are  equal,  =  A;  V  =  ^  X  SA  =  hA. 

Fora  wedge  with  parallel  ends,  42  =  0,  Am=--  \  Xi;V=|(4i+2A:)=-  —  • 

For  a  cone  or  pyramid,  Az  =  0,  Am  =  -  AI;  V  =  -  (A\  +  A\)  =  -^-- 

The  prismoidal  formula  is  a  rigid  formula  for  all  prismoids.  The  only 
approximation  involved  in  its  use  is  in  the  assumption  that  the  given  solid 
may  be  generated  by  a  right  line  moving  over  the  boundaries  of  the  end 
areas. 

The  area  of  the  middle  section  is  never  the  mean  of  the  two  end  areas  if 
the  prismoid  contains  any  pyramids  or  cones  among  its  elementary  forms. 
When  the  three  sections  are  similar  in  form  the  dimensions  of  the  middle 
area  are  always  the  means  of  the  corresponding  end  dimensions.  This 
fact  often  enables  the  dimensions,  and  hence  the  area  of  the  middle  section, 
to  be  computed  from  the  end  areas. 

Polyedrons.  —  A  polyedron  is  a  solid  bounded  by  plane  polygons.  A 
regular  polyedron  is  one  whose  sides  are  all  equal  regular  polygons. 

To  find  the  surface  of  a  regular  polyedron.  —  Multiply  the  area  of  one  of 
the  faces  by  the  number  of  faces;  9r,  multiply  the  square  of  one  of  the 
edges  by  the  surface  of  a  similar  solid  whose  edge  is  unity. 


A  TABLE  OP  THE'  REGULAR  POLYEDRONS  WHOSE  EDGES  ARE  UNITY. 

Names.                                  No*,  of  Faces.          Surface.  Volume. 

Tetraedron 4  1.7320508  0.1178513 

Hexaedron 6  6.0000000  1.0000000 

Octaedron 8  3.4641016  0.4714045 

Dodecaedron 12  20.6457288  7.6631189 

Icosaedroa 20  8.6602540  2.1816950 


g4  MENSURATION. 

To  find  the  volume  of  a  regular  polyedron.  —  Multiply  the  surface 
by  one  third  of  the  perpendicular  let  fall  from  the  centre  on  one  of  the 
faces;  or,  multiply  the  cube  of  one  of  the  edges  by  the  solidity  of  a  similar 
polyedron  whose  edge  is  unity. 

Solid  of  revolution.  —  The  volume  of  any  solid  of  revolution  is  equal 
to  the  product  of  the  area  of  its  generating  surface  by  the  length  of  the 
path  of  the  centre  of  gravity  of  that  surface. 

The  convex  surface  of  any  solid  of  revolution  is  equal  to  the  product  of 
the  perimeter  of  its  generating  surface  by  the  length  of  path  of  its  centre 
of  gravity. 

Cylindrical  ring.  —  Let  d  =  outer  diameter;  d'  =  inner  diameter; 
1/2  (d  -  d')  =  thickness  =  t;  1/4* I2  =  sectional  area;  1/2 (d  +d')  =  mean 
diameter  =  M;  m  =  circumference  of  section;  IT  M  =  mean  circum- 
ference of  ring;  surface  =  n  t  X  n  M;  =  1/4  ^  (d2  -  d/2);  =  9.86965  t  M ; 
=  2.46741  (d2  -  d/2);  volume  =  1/4  *  tz  M  n\  =  2.467241  .2  M. 

Spherical  zone.  —  Surface  of  a  spherical  zone,  or  segment  of  a  sphere 
=  its  altitude  X  the  circumference  of  a  great  circle  of  the  sphere.  A 
great  circle  is  one  v/hose  plane  passes  through  the  centre  of  the  sphere. 

Volume  of  a  zone  of  a  sphere.  —  To  the  sum  of  the  squares  of  the  radii 
of  the  ends  add  one  third  of  the  square  of  the  height;  multiply  the  sum 
by  the  height  and  by  1.5708. 

Spherical  segment.  —  Volume  of  a  spherical  segment  with  one  base.  — 
Multiply  half  the  height  of  the  segment  by  the  area  of  the  base,  and  the 
cube  of  the  height  by  0.5236  and  add  the  two  products.  Or,  from  three 
times  the  diameter  of  the  sphere  subtract  twice  the  height  of  the  segment; 
multiply  the  difference  by  the  square  of  the  height  and  by  0.5236.  Or,  to 
three  times  the  square  of  the  radius  of  the  base  of  the  segment  add  the 
square  of  its  height,  and  multiply  the  sum  by  the  height  and  by  0.5236. 

Spheroid  or  ellipsoid.  —  When  the  revolution  of  the  generating  sur- 
face of  the  spheroid  is  about  the  transverse  diameter  the  spheroid  is 
prolate,  and  when  about  the  conjugate  it  is  oblate. 

Convex  surface  of  a  segment  of  a  spheroid.  —  Square  the  diameters  of  the 
spheroid,  and  take  the  square  root  of  half  their  sum;  then,  as  the  diameter 
from  which  the  segment  is  cut  is  to  this  root  so  is  the  height  of  the  segment 
to  the  proportionate  height  of  the  segment  to  the  mean  diameter.  Multiply 
the  product  of  the  other  diameter  and  3. 1416  by  the  proportionate  height. 

Convex  surface  of  a  frustum  or  zone  of  a  spheroid.  —  Proceed  as  by 
previous  rule  for  the  surface  of  a  segment,  and  obtain  the  proportionate 
height  of  the  frustum.  Multiply  the  product  of  the  diameter  parallel  to 
the  base  of  the  frustum  and  3.1416  by  the  proportionate  height  of  the 
frustum. 

Volume  of  a  spheroid  is  equal  to  the  product  of  the  square  of  the  revol  v- 
ing  axis  by  the  fixed  axis  and  by  0.5236.  The  volume  of  a  spheroid  is  two 
thirds  of  that  of  the  circumscribing  cylinder. 

Volume  of  a  segment  of  a  spheroid.  —  1.  When  the  base  is  parallel  to  the 
revolving  axis,  multiply  the  difference  between  three  times  the  fixed  axis 
and  twice  the  height  of  the  segment,  by  the  square  of  the  height  and  by 
0.5236.  Multiply  the  product  by  the  square  of  the  revolving  axis,  and 
divide  by  the  square  of  the  fixed  axis. 

2.  When  the  base  is  perpendicular  to  the  revolving  axis,  multiply  the 
difference  between  three  times  the  revolving  axis  and  twice  the  height  of 
the  segment  by  the  square  of  the  height  and  by  0.5236.  Multiply  the 
product  by  the  length  of  the  fixed  axis,  and  divide  by  the  length  of  the 
revolving  axis. 

Volume  of  the  middle  frustum  of  a  spheroid.  —  1.  When  the  ends  are 
circular,  or  parallel  to  the  revolving  axis:  To  twice  the  square  of  the  middle 
diameter  add  the  square  of  the  diameter  of  one  end;  multiply  the  sum  by 
the  length  of  the  frustum  and  by  0.2618. 

2.  When  the  ends  are  elliptical,  or  perpendicular  to  the  revolving  axis: 
To  twice  the  product  of  the  transverse  and  conjugate  diameters  of  the 
middle  section  add  the  product  of  the  transverse  and  conjugate  diameters 
of  one  end;  multiply  the  sum  by  the  length  of  the  frustum  and  by  0.2618. 

Spindles.  —  Figures  generated  by  the  revolution  of  a  plane  area, 
bounded  by  a  ctirve  other  than  a  circle,  when  th  j  curve  is  revolved  about 
a  chord  perpendicular  to  its  axis,  or  about  its  double  ordinate.  They  are 
designated  by  the  name  of  the  arc  or  curve  from  which  they  are  generated, 
as  Circular,  Elliptic,  Parabolic,  etc.,  etc. 


MENSURATION.  65 

Convex  surface  of  a  circular  spindle,  zone,  or  segment  of  it.  —  Rule:  Mul- 
tiply the  length  by  the  radius  of  the  revolving  arc;  multiply  this  arc  by  the 
central  distance,  or  distance  between  the  centre  of  the  spindle  and  centre 
of  the  revolving  arc;  subtract  this  product  from  the  former,  double  the 
remainder,  and  multiply  it  by  3.1416. 

Volume  of  a  circular  spindle.  —  Multiply  the  central  distance  by  half 
the  area  of  the  revolving  segment;  subtract  the  product  from  one  third  of 
the  cube  of  half  the  length,  and  multiply  the  remainder  by  12.5664. 

Volume  of  fruslum  or  zone  of  a  circular  spindle.  —  From  the  square  of 
half  the  length  of  the  whole  spindle  take  one  third  of  the  square  of  half  the 
length  of  the  frustum,  and  multiply  the  remainder  by  the  said  half  length 
of  the  frustum;  multiply  the  central  distance  by  the  revolving  area  which 
generates  the  frustum;  subtract  this  product  from  the  former,  and  multi- 
ply the  remainder  by  6.2832. 

Volume  of  a  segment  of  a  circular  spindle.  —  Subtract  the  length  of  the 
segment  from  the  half  length  of  the  spindle;  double  the  remainder  and 
ascertain  the  volume  of  a  middle  frustum  of  this  length;  subtract  the 
result  from  the  volume  of  the  whole  spindle  and  halve  the  remainder. 


this  product  by  8. 

Parabolic  conoid.  —  Volume  of  a  parabolic  conoid  (generated  by  the 
revolution  of  a  parabola  on  its  axis).  —  Multiply  the  area  of  the  base  by 
half  the  height. 

Or  multiply  the  square  of  the  diameter  of  the  base  by  the  height  and  by 

Volume  of  a  fruslum  of  a  parabolic  conoid.  —  Multiply  half  the  sum  of 
xne  areas  of  the  two  ends  by  the  height. 

Volume  of  a  -parabolic  spindle  (generated  by  the  revolution  of  a  parabola 
on  its  base).  —  Multiply  the  square  of  the  middle  diameter  by  the  length 
and  by  0.4189.  The  volume  of  a  parabolic  spindle  is  to  that  of  a  cylinder 
of  the  same  height  and  diameter  as  8  to  15. 

Volume  of  the  middle  frustum  of  a  parabolic  spindle.  —  Add  together 
8  times  the  square  of  the  maximum  diameter,  3  times  the  square  of  the 
end  diameter,  and  4  times  the  product  of  the  diameters.  Multiply  the 
sum  by  the  length  of  the  frustum  and  by  0.05236.  This  rule  is  applicable 
for  calculating  the  content  of  casks  of  parabolic  form. 

Casks.  —  To  find  the  volume  of  a  cask  of  any  form.  —  Add  together  39 
times  the  square  of  the  bung  diameter,  25  times  the  square  of  the  head 
diameter,  and  26  times  the  product  of  the  diameters.  Multiply  the  sum 
by  the  length,  and  divide  by  31,773  for  the  content  in  Imperial  gallons,  or 
by  26,470  for  U.  S.  gallons. 

This  rule  was  framed  by  Dr.  Hutton,  on  the  supposition  that  the  middle 
third  of  the  length  of  the  cask  was  a  frustum  of  a  parabolic  spindle,  and 
each  outer  third  was  a  frustum  of  a  cone. 

To  find  the  ullage  of  a  cask,  the  quantity  of  liquor  in  it  when  it  is  not  full. 
1.  For  a  lying  cask:  Divide  the  number  of  wet  or  dry  inches  by  the  bung 
diameter  in  inches.  If  the  quotient  is  less  than  0.5,  deduct  from  it  one 
fourth  part  of  what  it  wants  of  0.5.  If  it  exceeds  0.5,  add  to  it  one  fourth 
part  of  the  excess  above  0.5.  Multiply  the  remainder  or  the  sum  by  the 
whole  content  of  the  cask.  The  product  is  the  quantity  of  liquor  in  the 
cask,  in  gallons,  when  the  dividend  is  wet  inches;  or  the  empty  space,  if 
dry  inches. 

2.  For  a  standing  cask:  Divide  the  number  of  wet  or  dry  inches  by  the 
length  of  the  cask.  If  the  quotient  exceeds  0.5,  add  to  it  one  tenth  of  its 
excess  above  0.5;  if  less  than  0.5,  subtract  from  it  one  tenth  of  what  it 
wants  of  0.5.  Multiply  the  sum  or  the  remainder  by  the  whole  content  of 
the  cask.  The  product  is  the  quantity  of  liquor  in  the  cask,  when  the 
dividend  is  wet  inches;  or  the  empty  space,  if  dry  inches. 

Volume  of  cask  (approximate)  U.  S.  gallons  =  square  of  mean  diam. 
X  length  in  inches  X  0.0034.  Mean  diameter  =  half  the  sum  of  the 
bung  and  head  diameters. 

Volume  of  an  irregular  solid.  —  Suppose  it  divided  into  parts,  resem- 
bling prisms  or  other  bodies  measurable  by  preceding  rules.  Find  the  con- 
lent  of  each  part;  the  sum  of  the  contents  is  the  cubic  contents  of  the  solid. 


66  PLANE   TRIGONOMETRY. 


The  content  of  a  small  part  is  found  nearly  by  multiplying  half  the  sum 
of  the  areas  of  each  end  by  the  perpendicular  distance  between  them. 

The  contents  of  small  irregular  solids  may  sometimes  be  found  by  im- 
mersing them  under  water  in  a  prismatic  or  cylindrical  vessel,  and  observ- 
ing the  amount  by  which  the  level  of  the  water  descends  when  the  solid  is 
withdrawn.  The  sectional  area  of  the  vessel  being  multiplied  by  the 
descent  of  the  level  gives  the  cubic  contents. 

Or,  weigh  the  solid  in  air  and  in  water;  the  difference  is  the  weight  of 
water  it  displaces.  Divide  the  weight  in  pounds  by  62.4  to  obtain  volume 
in  cubic  feet,  or  multiply  it  by  27.7  to  obtain  the  volume  in  cubic  inches. 

When  the  solid  is  very  large  and  a  great  degree  of  accuracy  is  not 
requisite,  measure  its  length,  breadth,  and  depth  in  several  different 
places,  and  take  the  mean  of  the  measurement  for  each  dimension,  and 
multiply  the  three  means  together. 

When  the  surface  of  the  solid  is  very  extensive  it  is  better  to  divide  it 
into  triangles,  to  find  the  area  of  each  triangle,  and  to  multiply  it  by  the 
mean  depth  of  the  triangle  for  the  contents  of  each  triangular  portion;  the 
contents  of  the  triangular  sections  are  to  be  added  together. 

The  mean  depth  of  a  triangular  section  is  obtained  by  measuring  the 
depth  at  each  angle,  adding  together  the  three  measurements,  and  taking 
one  third  of  the  sum. 


PLANE  TRIGONOMETRY. 

Trigonometrical  Functions. 

Every  triangle  has  six  parts  —  three  angles  and  three  sides.  When  any 
three  of  these  parts  are  given,  provided  one  of  them  is  a  side,  the  other 
parts  may  be  determined.  By  the  solution  of  a  triangle  is  meant  the 
determination  of  the  unknown  parts  of  a  triangle  when  certain  parts  are 
given. 

The  complement  of  an  angle  or  arc  is  what  remains  after  subtracting  the 
angle  or  arc  from  90°. 

In  general,  if  we  represent  any  arc  by  A,  its  complement  is  90°  -  A. 
Hence  the  complement  of  an  arc  that  exceeds  90°  is  negative. 

The  supplement  of  an  angle  or  arc  is  what  remains  after  subtracting  the 
angle  or  arc  from  180°.  If  A  is  an  arc  its  supplement  is  180°  —  A.  The 
supplement  of  an  arc  that  exceeds  180°  is  negative. 

The  sum  of  the  three  angles  of  a  triangle  is  equal  to  ISO0.  Either  angle  is 
the  supplement  of  the  other  two.  In  a  right-angled  triangle,  the  right 
angle  being  equal  to  90°,  each  of  the  acute  angles  is  the  complement  of 
the  other. 

In  all  right-angled  triangles  having  the  same  acute  angle,  the  sides  have  to 
each  other  the  same  ratio.  These  ratios  have  received  special  names,  as 
follows: 

If  A  is  one  of  the  acute  angles,  a  the  opposite  side,  b  the  adjacent  side, 
and  c  the  hypothenuse. 

The  sine  of  the  angle  A  is  the  quotient  of  the  opposite  side  divided  by  the 

hypothenuse.     Sin  A  == -• 

The  tangent  of  the  angle  A  is  the  quotient  of  the  opposite  side  divided  by 
the  adjacent  side.  Tan  A  =  j-- 

The  secant  of  the  angle  A  is  the  quotient  of  the  hypothenuse  divided  by  the 
adjacent  side.  Sec  A  =  -r  • 

The  cosine  (cos),  cotangent  (cot),  and  cosecant  (coscc)  of  an  angle 
are  respectively  the  sine,  tangent,  and  secant  of  the  complement  of  that 
angle.  The  terms  sine,  cosine,  etc.,  are  called  trigonometrical  functions. 

In  a  circle  whose  radius  is  unity,  the  sine  of  an  arc,  or  of  the  angle  at  the 
centre  measured  by  that  arc,  is  the  perpendicular  let  fall  from  one  extremity  of 
the  arc  upon  the  diameter  passing  through  the  other  extremity. 

The  tangent  of  an  arc  is  the  line  which  touches  the  circle  at  one  extremity 


PLANE   TRIGONOMETRY. 


67 


of  the  arc,  and  is  limited  by  the  diameter  (produced)  passing  through  the  other 
extremity. 

The  secant  of  an  arc  is  that  part  of  the  produced  diameter  which  is  inter" 
cepted  between  the  centre  and  the  tangent. 

The  versed  sine  of  an  arc  is  that  part  of  the  diameter  intercepted  between 
the  extremity  of  the  arc  and  the  foot  of  the  sine. 

In  a  circle  whose  radius  is  not  unity,  the  trigonometric  functions  of  an 
arc  will  be  equal  to  the  lines  here  denned,  divided  by  the  radius  of  the 
circle. 

it  1C  A  (Fig.  71)  is  an  angle  in  the  first  quadrant,  and  CF  =  radius, 


The  sine  of  the  angle  = 


FG 
Rad 


Cos  = 


Tan 


I A 

''  Had  ' 

Cosec  = 


Secant 

CL 
Rad  ' 


CT 

Rad  ' 

Versin  = 


CG 

Rad 

Cot  = 

GA 

''  Rad  * 


= 

Rad* 
PL 
Rad* 


FIG. 


If  radius  is  1,  then  Rad  in  the  denominator  is 
omitted,  and  sine  =  F  G,  etc. 

The  sine  of  an  arc  =  half  the  chord  of  twice  the 
arc. 

The  sine  of  the  supplement  of  the  arc  is  the 
same  as  that  of  the  arc  itself.     Sine  of  arc  B  D  F 
=  F  G  =  sin  arc  F  A. 
The  tangent  of  the  supplement  is  equal  to  the  tangent  of  the  arc,  but 
with  a  contrary  sign.     Tan  BDF  =  —  BM. 

The  secant  of  the  supplement  is  equal  to  the  secant  of  the  arc,  but  with 
a  contrary  sign.  Sec  BDF  =  —  CM. 

Signs  of  the  functions  in  the  four  quadrants.  —  If  we  divide  a 
circle  into  four  quadrants  by  a  vertical  and  a  horizontal  diameter,  the 
upper  right-hand  quadrant  is  called  the  first,  the  upper  left  the  second, 
the  lower  left  the  third,  and  the  lower  right  the  fourth.  The  signs  of  the 
functions  in  the  four  quadrants  are  as  follows: 

First  quad.  Second  quad.  Third  quad.  Fourth  quad. 
Sine  and  cosecant,  +  +  —  — 

Cosine  and  secant,  -4-  —  —  + 

Tangent  and  cotangent,    4-  —  +  — 

The  values  of  the  functions  are  as  follows  for  the  angles  specified: 


Angle  

o 

30 

45 

60 

QO 

120 

135 

150 

180 

970 

S60 

Sine  

0 

1 
2 

1 

V2 

v/3 
2 

1 

T~ 

1 

1 
2 

0 

-1 

0 

X/o 

I 

1 

1 

1 

\/^~ 

Cosine  

1 

~2 

V~2 

2* 

U 

2" 

2~ 

-1 

0 

1 

Tangent  

0 

J_ 

1 

Vs 

00 

-V3~ 

-1 

1 

0 

GO 

0 

Cotangent  .... 

00 

vf 

1 

I 

0 

J_ 

-1 

-\/3~3 

oo 

0 

\/3 

x/3 

Secant  

1 

2 

X/2 

2 

oo 

-2 

_x/2~ 

2 

-1 

00 

1 

Cosecant  

oc 

2 

\/2 

2 

v/3 

1 

2 

v? 

2 

oo 

-1 

to 

Versed  sine  ... 

d 

2-\/3 

\/2  i 

1 
2 

1 

3 

2 

V/J-f-l 

2+Va 

2 

1 

0 

2 

V2 

V2 

2 

68  PLANE  TRIGONOMETRY. 


TRIGONOMETRICAL,  FORMULAE. 

The  following  relations  are  deduced  from  the  properties  of  similai 
triangles  (Radius  =  1): 

cos  A  :  sin  A  : :  1  :  tan  A,  whence  tan  A  — r ; 

cos  A 

sin  A  :  cos  A  : :  1  :  cot  A.        "  cotan  A  =  —. — 7  ; 

sin  A 

cos  Ail         nl  i  sec  A,        "        sec  A 


cos  A' 

sin  A  1 1          : :  1  :  cosec  A,    "    cosec  A  —  -: — 7- ; 

sin  A 

tan  A  1 1  .       1 1 1  i  cot  A         ••      tan  A  =      1 


cot  A 

The  sum  of  the  square  of  the  sine  of  an  arc  and  the  square  of  its  cosine 
equals  unity.     Sin2  A  4-  cos2  A  =  1. 

Also,  1  4-  tan2  A  =  sec2  A;     I  +  cot2  A  =  cosec2  A. 

Functions  of  the  sum  and  difference  of  two  angles : 

Let  the  two  angles  be  denoted  by  A  and  B,  their  sum  A  4-  B  =*  C,  and 
their  difference  A  -  B  by  D. 

sin   (A  +  B)  =  sin  A  cos  B  4-  cos  A  sin  B; (1) 

cos  (A  +  B)  =  cos  A  cos  B  —  sin  A  sin  B; (2) 

sin  (A  —.  B)  =  sin  A  cos  B  —  cos  A  sin  B; (3) 

cos  (A  —  B)  =  cos  A  cos  B  +  sin  A  sin  B (4) 

From  these  four  formulae  by  addition  and  subtraction  we  obtain 

sin  (A  +  B)  +  sin  (A  -  B)  =  2  sin  A  cos  B;  .    .    .    .  (5 

sin  (A  +  B)  —  sin  (A  —  B)  =±  2  cos  A  sin  B;  .    .    .    .  (6 

cos  (A  +  B)  +  cos  (A  —  B)  =  2  cos  A  cos  5;  .    .    .    .  (7 

cos  (A  —  B)  —  cos  (A  4-  B)  =  2  sin  A  sin  5 (8 

If  we  put  A  +  B  =  C,  and  A  —  B  =  Z>,  then  A  =  1/2  (C  4-  D)  and  5  = 
v   and  we  have 

sin  (7  +  sin  D  =  2  sin  1/2(C  4-  D)  cos  i/2«?  -  D);  .  (9) 

sin  C  -  sin  D  =  2  cos  1/2  (C  4-  D)  sin  1/2  (C7  -  Z>);  .  .  (10) 

cos  C  +  cos  Z>=  20031/2(0  4-  D)  cos  i/2 ((7  -  D);  .  .  (11) 

cos  D  -  cos  C  =  2  sin  1/2  (C  4-  Z>)  sin  V2  (C  -  Z>).  .  .  (12) 

Equation  (9)  may  be  enunciated  thus:  The  sum  of  the  sines  of  any  two 
angles  is  equal  to  twice  the  sine  of  half  the  sum  of  the  angles  multiplied  by 
the  cosine  9f  half  their  difference.  These  formulae  enable  us  to  transform 
a  sum  or  difference  into  a  product. 

The  sum  of  the  sines  of  two  angles  is  to  their  difference  as  the  tangent  of 
half  the  sum  of  those  angles  is  to  the  tangent  of  half  their  difference. 

sin  A  4-  sin  B  =  2  sin  V2(A  4-  B)  cos  V2(A  -B)        tan  V2  (A  4-  B} 
sin  A  -  sin  B       2  cos  i/2  (A  +  B)  sin  i/2  (A  -  B)  **"  tan  i/2  (A  -  B)' 

The  sum  of  the  cosines  of  two  angles  is  to  their  difference  as  the  cotan- 
gent.of  half  the  sum  of  those  angles  is  to  the  tangent  of  half  their  difference. 

cos  A  4-  cos  B  =  2  cos  l/2(A  4-  B}  cos  V2(A  -B)  =   cot  l/2(A4-£)[      (     . 
cos  B  -  cos  A       2  sin  1/2  (A  4-  B)  sin  1/2  (A  -  B)      tan  i/2  (A  -  B) ' 

The  sine  of  the  sum  of  two  angles  is  to  the  sine  of  their  difference  as  the 
sum  of  the  tangents  of  those  angles  is  to  the  difference  of  the  tangents. 


sin  (A  4-  B)  ^  tan  A  +  tan  B . 
sin  (A  -  £)       tan  A  -  tan  B ' 


(15) 


PLANE   TRK 
MnU+A)              !            jj. 

3ONOMET 

tan  (A-f 
tan  (A  — 
cot   (A  + 
cot  (A  — 

cos  2  A 
cot  2A 

cos  1/2  A 
cot  1/2  A 

BY.                              69 

£.      tan  A  -f  tan  3  . 

cos  A  cos  5 

sin  (A  —  5) 

P         tan  A  -  tan  B  . 

cosAcosl?"  ^                        •*' 
cos  (A  4-  B)                     itanJB- 

••       1  +  tan  A  tan  ,6  * 

cos  A  cos  5 
cos  (A  —  J5)      t 

cot  B  +  cot  A  ' 

cos  A  cos  5 
Functions  of  twice  an  angle: 

sin'  2  A  =  2  sin  A  cos  A; 
tin  01           2  tan  A 

cot  B  —  cot  A 

«=  cos2  A  —  sin2  A  ; 
cot2  A  -  1 

~  1  -  tan2  A  * 
Functions  of  half  an  angle: 

2  cot  A 

.  /  1  —  cos  A 

•    J  1  +  cos  A. 

cm  1/2  A-  -J.  y         2         ; 

!a*-L     V           2 
\/l  4-  cos  A 

tin  I/*  1         f    i/1    ~  C°S  A    - 

»    1  4-  cos  A    ' 

V  i  —  cos  A 

For  tables  of  Trigonometric  Functions,  see  Mathematical  Tables. 


Solution  of  Plane  Right-angled  Triangles. 


Let  A  and  B  be  the  two  acute  angles  and  C  the  right  angle,  and  a,  6,  and 
c  the  sides  opposite  these  angles,  respectively,  then  we  have 

d  " 

1.   sin  A  =  cos  B  =  ~  ;      3.   tan  A 


2.   cos  A  =  sin  £ 


4.   cot  A  =  tan  B 


1.  In  any  plane  right-angled  triangle  the  sine  of  either  of  the  acute 
angles  is  equal  to  the  quotient  of  the  opposite  leg  divided  by  the  hypothe- 
nuse. 

2.  The  cosine  of  either  of  the  acute  angles  is  equal  to  the  quotient  of 
the  adjacent  leg  divided  by  the  hypothenuse. 

3.  The  tangent  of  either  of  the  acute  angles  is  equal  to  the  quotient  of 
the  opposite  leg  divided  by  .the  adjacent  leg. 

4.  The  cotangent  of  either  of  the  acute  angles  is  equal  to  the  quotient 
of  the  adjacent  Teg  divided  by  the  opposite  leg. 

5.  The  square  of  the  hypothenuse  equals  the  sum  of  the  squares  of  the 
other  two  sides. 


Solution  of  Oblique-angled  Triangles. 

The  following  propositions  are  proved  in  works  on  plane  trigonometry. 
In  any  plane  triangle  — • 

Theorem  1.  The  sines  of  the  angles  are  proportional  to  the  opposite 
sides. 

Theorem  2.  The' sum  of  any  two  sides  is  to  their  difference  as  the  tan- 
gent of  half  the  sum  of  the  opposite  angles  is  to  the  tangent  of  half  their 
difference. 

Theorem  3.  If  from  any  angle  of  a  triangle  a  perpendicular  be  drawn  to 
the  opposite  side  or  base,  the  whole  base  will  be  to  the  sum  of  the  other 
two  sides  as  the  difference  of  those  two  sides  is  to  the  difference  of  the 
segments  of  the  base. 

CASE  I.  Given  two  angles  and  a  side,  to  find  the  third  angle  and  the 
other  two  sides.  1.  The  third  angle  —  180°  —  sum  of  the  two  angles. 
2.  The  sides  may  be  found  by  tlie  following  proportion; 


70  ANALYTICAL  GEOMETRY. 

The  sine  of  the  angle  opposite  the  given  side  is  to  the  sine  of  the  angle 
opposite  the  required  side  as  the  given  side  is  to  the  required  side. 

CASE  II.  Given  two  sides  and  an  angle  opposite  one  of  them,  to  find 
the  third  side  and  the  remaining  angles. 

The  side  opposite  the  given  angle  is  to  the  side  opposite  the  required 
angle  as  the  sine  of  the  given  angle  is  to  the  sine  of  the  required  angle. 

The  third  angle  is  found  by  subtracting  the  sum  of  the  other  two  from 
180°,  and  the  third  side  is  found  as  in  Case  I. 

CASE  III.  Given  two  sides  and  the  included  angle,  to  find  the  third 
side  and  the  remaining  angles. 

The  sum  of  the  required  angles  is  found  by  subtracting  the  given  angle 
from  180°.  The  difference  of  the  required  angles  is  then  found  by  Theorem 
II.  Half  the  difference  added  to  half  the  sum  gives  the  greater  angle,  and 
half  the  difference  subtracted  from  half  the  sum  gives  the  less  angle.  The 
third  side  is  then  found  by  Theorem  I. 

Another  method: 

Given  the  sides  c,  6,  and  the  included  angle  A,  to  find  the  remaining  side 
a  and  the  remaining  angles  B  and  C. 

From  either  of  the  unknown  angles,  as  B,  draw  a  perpendicular  Be  to 
the  opposite  side. 

Then 

Ae  =  c  cos  A,     Be  =  c  sin  A,    eC  =  b  —  Ac     Be  •*•  eC  =  tan  C. 

Or,  in  other  words,  solve  Be,  Ae  and  BeC  as  right-angled  triangles. 

CASE  IV.    Given  the  three  sides,  to  find  the  angles. 

Let  fall  a  perpendicular  upon  the  longest  side  from  the  opposite  angle, 
dividing  the  given  triangle  into  two  right-angled  triangles.  The  two  seg- 
ments of  the  base  may  be  found  by  Theorem  III.  There  will  then  be 
given  the  hypothenuse  and  one  side  of  a  right-angled  triangle  to  find  the 
angles. 

For  areas  of  triangles,  see  Mensuration. 


ANALYTICAL  GEOMETRY. 

Analytical  geometry  is  that  branch  of  Mathematics  which  has  for  its 
object  the  determination  of  the  forms  and  magnitudes  of  geometrical 
magnitudes  by  means  of  analysis. 

Ordinates  and  abscissas.  —  In  analytical  geometry  two  intersecting 
lines  YY',  XX'  are  used  as  coordinate  axes, 
XX'  being  the  axis  of  abscissas  or  axis  of  X, 
and  YY'  the  axis  of  ordinates  or  axis  of  Y. 

A,  the  intersection,  is  called  the  origin  of  co-  /:; 7 

ordinates.     The  distance  of   any   point   P  /u         / 

from  the  axis  of  Y  measured  parallel  to  the  / 

axis  of  X  is  called  the  abscissa  of  the  point, 
as  AD  or  CP,  Fig.  72.     Its  distance  from  the 


f 

V' 


axis  of  X,  measured  parallel  to  the  axis  of 

Y,  is  called  the  ordinate,   as  AC  or  PD. 

The  abscissa  and  ordinate  taken  together 

are  called  the  coordinates  of  the  point  P. 

The  angle  of  intersection  is  usually  taken  as  Y 

a  right  angle,  in  which  case  the  axes  of  X  pIG    72 

and  Y  are  called  rectangular  coordinates. 

The  abscissa  of  a  point  is  designated  by  the  letter  x  and  the  ordinate 
oy  y. 

The  equations  of  a  point  are  the  equations  which  express  the  distances 
of  the  point  from  the  axis.  Thus  x  =  a,  y  =  b  are  the  equations  of  the 
point  P. 

Equations  referred  to  rectangular  coordinates.  —  The  equation  of 
a  line  expresses  the  relation  which  exists  between  the  coordinates  of  every 
point  of  the  line. 

Equation  of  a  straight  line,  y  =  ax  ±  b,  in  which  a  is  the  tangent  of  the 
angle  the  line  makes  with  the  axis  of  -Y,  and  b  the  distance  above  A  in 
which  the  line  cuts  the  axis  of  Y. 

Every  equation  of  the  first  degree  between  two  variables  is  the  equation 


ANALYTICAL    GEOMETRY.  71 

of  a  straight  line,  as  Ay  4-  Bx  f  C  »  0,  which  can  be  reduced  to  the  form 
y  =  o#  ±  6. 

Equation  of  the  distance  between  two  points: 

D  =  vV'  -  z')2  +  (y"  -  I/O2, 

in  which  x'y',  x"y"  are  the  coordinates  of  the  two  points. 
Equation  of  a  line  passing  through  a  given  point: 

y  -  y'  =  a(x  -  x'), 

in  which  x'y'  are  the  coordinates  of  the  given  point,  a,  the  tangent  of  the 
angle  the  line  makes  with  the  axis  of  x,  being  undetermined,  since  any 
number  of  lines  may  be  drawn  through  a  given  point. 
Equation  of  a  line  passing  through  two  given  points: 


Equation  of  a  line  parallel  to  a  given  line  and  through  a  given  point: 

y  —  y'  =  a(x  —  x'}. 
Equation  of  an  angle  V  included  between  two  given  lines: 

a'  —  a 


in  which  a  and  a'  are  the  tangents  of  the  angles  the  lines  make  with  the 
axis  of  abscissas. 

If  the  lines  are  at  right  angles  to  each  other  tang  V  =  oo,  and 

1  +  a'a  =  0. 

Equations  of  an  intersection  of  two  lines,    whose  equations  are 
y  =  ax   f  b,         and    y  =  a'x  +•  &', 
b  -  b'  ab'  -  a'b 

x  -  ~  ^r-rf*   and   y  =  T^5T 

Equation  of  a  perpendicular  from  a  given  point  to  a  given  line: 

y  -  y'  =  -  -  (x*  -  x'). 
Equation  of  the  length  of  the  perpendicular  Pi 


The  circle.  —  Equation  of  a  circle,  the  origin  of  coordinates  being  at 
the  centre,  and  radius  -=  A': 

x2  -f  2/2  =  R*. 
II  the  origin  is  at  the  left  extremity  of  the  diameter,  on  the  axis  of  X: 

y2  =  2Rx  -  x2. 

If  the  origin  is  at  any  point,  and  the  coordinates  of  the  centre  are  x'y' 
(x  -  z')2  +  (y  -  2/')2  =  #2. 

Equation  of  a  tangent  to  a  circle,  the  coordinates  of  the  point  of  tan- 
gency  being  x"y"  and  the  origin  at  the  centre, 

yy"  +  xx"  =  R2. 

The  ellipse.  —  Equation  of  an  ellipse,  referred  to  rectangular  coordi- 
nates with  axis  at  the  centre: 

AW  +  £2x2  =  A*B\ 
in  which  4  is  half  tUe  transverse  axis  and  £  qajf  the  conjugate  **fs. 


72  ANALYTICAL    GEOMETRY. 

Equation  of  the  ellipse  wiien  the  origin  is  at  the  vertex  of  the  transverse 
axis; 

B2 
y*  =  ~j(2Ax  -  *'). 

The  eccentricity  of  an  ellipse  is  the  distance  from  the  centre  to  either 
focus,  divided  by  the  semi-transverse  axis,  or 


The  parameter  of  an  ellipse  is  the  double  ordinate  passing  through  the 
focus.  It  is  a  third  proportional  to  the  transverse  axis  and  its  conjugate, 
or 

2»2 

2  A  :  2B  ::  2B  :  parameter;  or  parameter  =  -^— 

Any  ordinate  of  a  circle  circumscribing  an  ellipse  is  to  the  corresponding 
ordinate  of  the  ellipse  as  the  semi  -trans  verse  axis  to  the  semi-conjugate. 
Any  ordinate  of  a  circle  inscribed  in  an  ellipse  is  to  the  corresponding 
ordinate  of  the  ellipse  as  the  semi  -conjugate  axis  to  the  semi-transverse. 

Equation  of  the  tangent  to  an  ellipse,  origin  of  axes  at  the  centre: 

A*yy"  +  Bzxx"  =  A*B*. 

y"x"  being  the  coordinates  of  the  point  of  tangency. 

Equation  of  the  normal,  passing  through  the  point  of  tangency,  and 
perpendicular  to  the  tangent: 

»-v-s5?<*-*">- 

The  normal  bisects  the  angle  of  the  two  lines  drawn  from  the  point  of 
tangency  to  the  foci. 

The  lines  drawn  from  the  foci  make  equal  angles  with  the  tangent. 

The  parabola.  —  Equation  of  the  parabola  referred  to  rectangular 
coordinates,  the  origin  being  at  the  vertex  of  its  axis,  y2  =  2px,  in  which 
2p  is  the  parameter  or  double  ordinate  through  the  focus. 

The  parameter  is  a  third  proportional  to  any  abscissa  and  its  correspond- 
ing ordinate,  or 

x  :  y  ::  y  :  2p. 

Equation  of  the  tangent: 

yy"  =  p(x 


y"x"  being  coordinates  of  the  point  of  tangency. 
Equation  of  the  normal: 

y  -  y"  -  -  ~(x  -  x"). 

The  sub-normal,  or  projection  of  the  normal  on  the  axis,  is  constant,  and 
equal  to  half  the  parameter. 

The  tangent  at  any  point  makes  equal  angles  with  the  axis  and  with  the 
line  drawn  from  the  pDint  of  tangency  to  the  focus. 

The  hyperbola.  —  Equation  of  the  hyperbola  referred  to  rectangular 
coordinates,  origin  at  the  centre: 


in  which  A  is  the  semi-transverse  axis  and  B  the  semi-conjugate  axis. 
Equation  when  the  origin  is  at  the  right  vertex  of  the  transverse  axis: 


Conjugate  and  equilateral  hyperbolas.  —  If  on  the  conjugate  axis 


DIFFERENTIAL   CALCULUS.  73 

as  a  transverse,  and  a  focal  distance  equal  to  ^A2  + Bz,  we  construct 
the  two  branches  of  a  hyperbola,  the  two  hyperbolas  thus  constructed  are 
called  conjugate  hyperbolas.  If  the  transverse  and  conjugate  axes  are 
equal,  the  hyperbolas  are  called  equilateral,  in  which  case  y*-x2=  -A* 
when  A  is  the  transverse  axis,  and  x2  -  ?/2  =  —  B2  when  B  is  the  trans- 

The  parameter  of  the  transverse  axis  is  a  third  proportional  to  the  trans- 
rerse  axis  and  its  conjugate. 

2 A  :  2B  ::  2J5  :  parameter. 

The  tangent  to  a  hyperbola  bisects  the  angle  of  the  two  lines  drawn  from 
the  point  of  tangency  to  the  foci. 

The  asymptotes  of  a  hyperbola  are  the  diagonals  of  the  rectangle 
described  on  the  axes,  indefinitely  produced  in  both  directions. 

The  asymptotes  continually  approach  the  hyperbola,  and  become 
tangent  to  it  "at  an  infinite  distance  from  the  centre. 

Equilateral  hyperbola.  —  In  an  equilateral  hyperbola  the  asymptotes 
make  equal  angles  with  the  transverse  axis,  and  are  at  right  angles  to  each 
other.  With  the  asymptotes  as  axes,  and  P  =  ordinate,  V  —  abscissa, 
py  =  a  constant.  This  equation  is  that  of  the  expansion  of  a  perfect 
gas,  in  which  P  =  absolute  pressure,  V  =  volume. 

Curveof  Expansion  of  Gases.  —  PV™  =  a  constant,  or  Pi  Vin=PzVzn, 
in  which  Fi  and  ¥2  are  the  volumes  at  the  pressures  Pi  and  Pz.  When 
these  are  given,  the  exponent  n  may  be  found  from  the  formula 


. 

1 


log  Pi  -  log  Pz 

log  Vz  —  log  Vi 


Conic  sections,  —  Every  equation  of  the  second  degree  between  two 
variables  will  represent  either  a  circle,  an  ellipse,  a  parabola  or  a  hyperbola. 
These  curves  are  those  which  are  obtained  by  intersecting  the  surface  of  a 
cone  by  planes,  and  for  this  reason  they  are  called  conic  sections. 

Logarithmic  curve,  —  A  logarithmic  curve  is  one  in  which  one  of  the 
coordinates  of  any  point  is  the  logarithm  of  the  other. 

The  coordinate  axis  to  which  the  lines  denoting  the  logarithms  are 
parallel  is  called  the  axis  of  logarithms,  and  the  other  the  axis  of  numbers. 
If  y  is  the  axis  of  logarithms  and  x  the  axis  of  numbers,  the  equation  of  the 
curve  is  y  =  log  x. 

If  the  base  of  a  system  of  logarithms  is  a,  we  have  ay  =  x,  in  which  y  is 
the  logarithm  of  x. 

Each  system  of  logarithms  will  give  a  different  logarithmic  curve.  If 
y  ^  o,  x  =  1.  Hence  every  logarithmic  curve  will  intersect  the  axis  of 
numbers  at  a  distance  from  the  origin  equal  to  1. 


DIFFERENTIAL  CALCULUS. 

The  differential  of  a  variable  quantity  is  the  difference  between  any  two 
of  its  consecutive  values;  hence  it  is  indefinitely  small.  It  is  expressed  by 
writing  d  before  the  quantity,  as  dx,  which  is  read  differential  of  x. 

The  term  ^  is  called  the  differential  coefficient  of  y  regarded  as  a  func- 
tion of  x.  It  is  also  called  the  first  derived  function  or  the  derivative. 

The  differential  of  a  function  is  equal  .to  its  differential  coefficient  mul- 
tiplied by  the  differential  of  the  independent  variable;  thus,  -^dx  =  dy. 

The  limit  of  a  variable  quantity  is  that  value  to  which  it  continually 
approaches,  so  as  at  last  to  differ  from  it  by  less  than  any  assignable 
quantity^ 

The  differential  coefficient  is  the  limit  of  the  ratio  of  the  increment  of 
the  independent  variable  to  the  increment  of  the  function. 

The  differential  of  a  constant  quantity  is  equal  to  0. 

The  differential  of  a  product  of  a  constant  by  a  variable  is  equal  to  the 
constant  multiplied  by  the  differential  of  the  variable. 

If  u  =  Av,     du  =  A  dv* 


74  DIFFERENTIAL  CALCULUS. 

In  any  curve  whose  equation  is  y  =  /(#),  the  differential  coefficient 
•5T-  =  tan  a;  hence,  the  rate  of  increase  of  the  function,  or  the  ascension  of 

the  curve  at  any  point,  is  equal  to  the  tangent  of  the  angle  which  the 
tangent  line  makes  with  the  axis  of  abscissas. 

All  the  operations  of  the  Differential  Calculus  comprise  but  two  objects: 

1.  To  find  the  rate  of  change  in  a  function  when  it  passes  from  one  state 
of  value  to  another,  consecutive  with  it. 

2.  To  find  the  actual  change  in  the  function:  The  rate  of  change  is  the 
differential  coefficient,  and  the  actual  change  the  differential. 

Differentials  of  algebraic  functions.  —  The  differential  of  the  sum 
or  difference  of  any  number  of  functions,  dependent  on  the  same  variable, 
is  equal  to  the  sum  or  difference  of  their  differentials  taken  separately: 

If    u  =  y  4-  z  —  w,    du  —  dy  +  dz  —  dw. 

The  differential  of  a  product  of  two  functions  dependent  on  the  same 
variable  is  equal  to  the  sum  of  the  products  of  each  by  the  differential  of 
the  other: 

_      74.     fj       d(uv)   _  du_       dv 
uv  u         v 

The  differential  of  the  product  of  any  number  ol  functions  is  equal  to 
the  sum  of  the  products  which  arise  by  multiplying  the  differential  of  each 
function  by  the  product  of  all  the  others: 

d(uts)  —  tsdu  +  usdt  +  utds. 

The  differential  of  a  fraction  equals  the  denominator  into  the  diffeiential 
of  the  numerator  minus  the  numerator  into  the  differential  of  the  denom- 
inator, divided  by  the  square  of  the  denominator: 

_      (tL\  —  v^u~  u  dv 

If  the  denominator  is  constant,  dv  =  0,  and  dt  —  — 5-  =  —  • 

v  v 

If  the  numerator  is  constant,  du  =  0,  and  dt  = -$• 

The  differential  of  the  square  root  of  a  quantity  is  equal  to  the  differen- 
tial of  the  quantity  divided  by  twice  the  square  root  of  the  quantity: 


If     v  =  it1/2'     or     v  - 


2V  u 


2 


The  differential  of  any  power  of  a  function  is  equal  to  the  exponent  multi- 
plied by  the  function  raised  to  a  powerless  one,  multiplied  by  the  differen- 
tial of  the  function,  d(un)  =  nun~ldu. 

Formulas  for  differentiating  algebraic  functions. 

1.  d  (a)  =  0. 

2.  d  (ax)  =  a  dx. 

3.  d  (x  +  y)  =  dx  +  dy. 

4.  d  (x  —  y)  =  dx  —  dy. 

5.  d  (xy)  =  x  dy  +  y  dx. 

To  find  the  differential  of  the  form  u  =  (a  +  bxn)m: 
Multiply  the  exponent  of  the  parenthesis  into  the  exponent  of  the  vari- 
able within  the  parenthesis,  into  the  coefficient  of  the  variable,  into  the 


DIFFERENTIAL   CALCULUS.  75 

binomial  raised  to  a  power  less  1 ,  into  the  variable  within  the  parenthesis 
raised  to  a  power  less  1,  into  the  differential  of  the  variable. 

du  =  d(a  +  bxn)m  =  mnb(a  +  bxn)m~l  xn~l  dx. 

To  find  the  rate  of  change  for  a  given  value  of  the  variable: 
Find  the  differential  coefficient,  and  substitute  the  value  of  the  variable 
in  the  second  member  of  the  equation. 

EXAMPLE.  —  If  x  is  the  side  of  a  cube  and  u  its  volume,  u  =  x3,  -r-  =  3x2. 

Hence  the  rate  of  change  in  the  volume  is  three  times  the  square  of  the 
edge.  If  the  edge  is  denoted  by  1,  the  rate  of  change  is  3. 

Application.  The  coefficient  of  expansion  by  heat  of  the  volume  of  a 
body  is  three  times  the  linear  coefficient  of  expansion.  Thus  if  the  side 
of  a  cube  expands  0.001  inch,  its  volume  expands  0.003  cubic  inch.  1.0013 
=  1.003003001. 

A  partial  differential  coefficient  is  the  differential  coefficient  of  a 
function  of  two  or  more  variables  under  the  supposition  that  only  one 
of  them  has  changed  its  value. 

A  partial  differential  is  the  differential  of  a  function  of  two  or  more 
variables  under  the  supposition  that  only  one  of  them  has  changed  its 
value. 

The  total  differential  of  a  function  of  any  number  of  variables  is  equal 
to  the  sum  of  the  partial  differentials. 

If  u  =  f  (xy),  the  partial  differentials  are  -r-  dx,  ~rdy. 

' 

Integrals.  —  An  integral  is  a  functional  'expression  derived  from  a 
differential.  Integration  is  the  operation  of  finding  the  primitive  func- 
tion from  the  differential  function.  It  is  indicated  by  the  sign/i  which  is 

read  "the  integral  of."  Thus  fix  dx  =  z2;  read,  the  integral  of  2xdx 
equals  x2. 

To  integrate  an  expression  of  the  form  mxm~1dx  or  xmdx,  add  1  to  the 
exponent  of  the  variable,  and  divide  by  the  new  exponent  and  by  the 

differential  of  the  variable:  JZx^dx  =  a:3.     (Applicable  in  all  cases  except 

when   m  =  —  1.     For  Jx      dx  see  formula  2,  page  81.) 

The  integral  of  the  product  of  a  constant  by  the  differential  of  a  vari- 
*)le  is  equal  to  the  constant  multiplied  by  the  integral  of  the  differential: 


If  u  -=  x*  +  y3  -  z,  du  =  -     dx  +       dy  +       dz;  =  2xdx  +  3y*  dy  -  dz. 


fax™  dx  =  a   f 


xmdx  =  a 


m  +  1* 


The  integral  of   the  algebraic  sum  of   any  number  of  differentials  is 
equal  to  the  algebraic  sum  of  their  integrals: 


du  =  2axzdx  —  bydy—  z2  dz;  (  du=  - 


Since  the  differential  of  a  constant  is  0,  a  constant  connected  with  a 
variable  by  the  sign  +  or  —  disappears  in  the  differentiation;  thus 
d(a  -4-  xm)  =  dxm  =  mxm~l  dx.  Hence  in  integrating  a  differential 
expression  we  must  annex  to  the  integral  obtained  a  constant  represented 
by  C  to  compensate  for  the  term  which  may  have  been  lost  in  differen- 
tiation. Thus  if  we  have  dy  =  adx^fdy  =»  afdx.  Integrating, 

y  =  ax  ±  C. 


76  DIFFERENTIAL   CALCULUS. 

The  constant  C,  which  is  added  to  the  first  integral,  must  have  such  a 
value  as  to  render  the  functional  equation  true  for  every  possible  value 
that  may  be  attributed  to  the  variable.  Hence,  after  having  found  the 
first  integral  equation  and  added  the  constant  C,  if  we  then  make 
the  variable  equal  to  zero,  the  value  which  the  function  assumes  will  be 
the  true  value  of  C. 

An  indefinite  integral  is  the  first  integral  obtained  before  the  value  of 
the  constant  C  is  determined. 

A  particular  integral  is  the  integral  after  the  value  of  C  has  been  found. 

A  definite  integral  is  the  integral  corresponding  to  a  given  value  of  the 
'-ariable. 

Integration  between  limits.  —  Having  found  the  indefinite  integral 
and  the  particular  integral,  the  next  step  is  to  find  the  definite  integral 
and  then  the  definite  integral  between  given  limits  of  the  variable. 

The  integral  of  a  function,  taken  between  two  limits,  indicated  by  given 
values  of  x,  is  equal  to  the  difference  of  the  definite  integrals  correspond- 
ing to  those  limits.  The  expression 


X 


X" 

dy 


is  read:  Integral  of  the  differential  of  y,  taken  between  the  limits  xf  and 
x"\  the  least  limit,  or  the  limit  corresponding  to  the  subtractive  integral, 
being  placed  below. 

Integrate  du  •—  9xz  dx  between  the  limits  x  =  1  and  x  =  3,  u  being  equal 

to  81  when  x  =  0.     /du  =  /Qxz  dx  =  3x3  -f  C;  C  =  81  when  x  =  0,  then 

=  3 

du  =  3(3)3  +  8i>  minus  3(1)3  +  »i  =  73. 


Integration  of  particular  forms. 

To  integrate  a  differential  of  the  form  du  =  (a  +  bxn)mxn    l  dx. 

1.  If  there  is  a  constant  factor,  place  it  without  the  sign  of  the  integral, 
and  omit  the  power  of  the  variable  without  the  parenthesis  and  the  differ- 
ential ; 

2.  Augment   the  exponent  of   the  parenthesis  by  1,  and  then  divide 
this  quantity,  with  the  exponent  so  increased,  by  the  exponent  of  the 
parenthesis,  into  the  exponent  of  the  variable  within  the  parenthesis, 
into  the  coefficient  of  the  variable.     Whence 


(wH-Dnd 

The  differential  of  an  arc  is  the  hypothenuse  of  a  right-angle  triangle  of 
which  the  base  is  dx  and  the  perpendicular  dy. 

If  2  is  an  arc,  dz  =  ^dxz  +  dyz    z  =J  ^dx2  +  dy*. 

Quadrature  of  a  plane  figure. 

The  differential  of  the  area  of  a  plane  surface  is  equal  to  the  ordmate  int^ 
the  differential  of  the  abscissa. 

ds  =  y  dx. 

To  apply  the  principle  enunciated  in  the  last  equation,  in  finding  the  area 
of  any  particular  plane  surface: 

Find  the  value  of  y  in  terms  of  x,  from  the  equation  of  the  bounding  line; 
substitute  this  value  in  the  differential  equation,  and  then  integrate 
between  the  required  limits  of  x. 

Area  of  the  parabola.  —  Find  the  area  of  any  portion  of  the  com- 
mon parabola  whose  equation  is 

yz  =  2px;     whence  y  = 


DIFFERENTIAL   CALCULUS.  77 

Substituting  this  value  of  y  in  the  differential  equation  ds  =  y  dx  gives 


If  we  estimate  the  area  from  the  principal  vertex,  x  =  0,  y  =  0,  and 

o 
C  =  0;  and  denoting  the  particular  integral  by  s7,  s'  =  ^  zi/. 

o 

That  is,  the  area  of  any  portion  of  the  parabola,  estimated  from  the 
vertex,  is  equal  to  2/3  of  the  rectangle  of  the  abscissa  and  ordinate  of  the 
extreme  point.  The  curve  is  therefore  quadrable. 

Quadrature  of  surfaces  of  revolution.  —  The  differential  of  a  surface 
of  revolution  is  equal  to  the  circumference  of  a  circle  perpendicular  to  the 
axis  into  the  differential  of  the  arc  of  the  meridian  curve. 


•  ds  = 

in  which  y  is  the  radius  of  a  circle  of  the  bounding  surface  in  a  i 
pendicular  to  the  axis  of  revolution,  and  r  is  the  abscissa,  or  distance  of 
the  plane  from  the  origin  of  coordinate  axes. 

Therefore,  to  find  the  volume  of  any  surface  of  revolution: 

Find  the  value  of  y  and  dy  from  the  equation  of  the  meridian  curve  in 
terms  of  x  and  dx,  then  substitute  these  values  in  the  differential  equation, 
and  integrate  between  the  proper  limits  of  x. 

By  application  of  this  rule  we  may  find: 

The  curved  surface  of  a  cylinder  equals  the  product  of  the  circum- 
ference of  the  base  into  the  altitude. 

The  convex  surface  of  a  cone  equals  the  product  of  the  circumference  of 
the  base  into  half  the  slant  height. 

The  surface  of  a  sphere  is  equal  to  the  area  of  four  great  circles,  or  equal 
to  the  curved  surface  of  the  circumscribing  cylinder. 

Cubature  of  volumes  of  revolution.  —  A  volume  of  revolution  is  a 
volume  generated  by  the  revolution  of  a  plane  figure  about  a  fixed  line 
called  the  axis. 

If  we  denote  the  volume  by  V,  dV  =  xy2  dx. 

The  area  of  a  circle  described  by  any  ordinate  y  is  ny2;  hence  the  differ- 
ential of  a  volume  of  revolution  is  equal  t9  the  area  of  a  circle  perpendicular 
to  the  axis  into  the  differential  of  the  axis. 

The  differential  of  a  volume  generated  by  the  revolution  of  a  plane 
figure  about  the  axis  of  Y  is  nx2  dy. 

To  find  the  value  of  V  for  any  given  volume  of  revolution  : 

Find  the  value  of  y2  in  terms  of  x  from  the  equation  of  the  meridian 
curve,  substitute  this  value  in  the  differential  equation,  and  then  integrate 
between  the  required  limits  of  x. 

By  application  of  this  rule  we  may  find: 

The  volume  of  a  cylinder  is  equal  to  the  area  of  the  base  multiplied 
by  the  altitude. 

The  volume  of  a  cone  is  equal  to  the  area  of  the  base  into  one  third  the 
altitude. 

The  volume  of  a  prolate  spheroid  and  of  an  oblate  spheroid  (formed  by 
the  revolution  of  an  ellipse  around  its  transverse  and  its  conjugate  axis 
respectively)  are  each  equal  to  two  thirds  of  the  circumscribing  cylinder. 

If  the  axes  are  equal,  the  spheroid  becomes  a  sphere  and  its  volume  = 

-  nRz  X  D  =  -  7rZ>3;  R  being  radius  and  D  diameter. 
o  o 

The  volume  of  a  paraboloid  is  equal  to  half  the  cylinder  having  the  same 
base  and  altitude. 

The  volume  of  a  pyramid  equals  the  area  of  the  base  multiplied  by  one 
third  the  altitude. 

Second,  third,  etc.,  differentials.  —  The  differential  coefficient  being 
a  function  of  the  independent  variable,  it  may  be  differentiated,  and  we 
thus  obtain  the  second  differential  coefficient; 


78  DIFFERENTIAL  CALCULUS 

^\  =-  ——•      Dividing  by  dxt  we  have  for  the  second  differential 


coefficient  -r-^,  which  is  read  :  second  differential  of  u  divided  by  the  square 
of  the  differential  of  x  (or  dx  squared). 

The  third  differential  coefficient   ^  is  read:  third  differential  of  u 

divided  by  dx  cubed. 

The  differentials  of  the  different  orders  are  obtained  by  multiplying 
the  differential  coefficient  by  the  corresponding  powers  of  dx;  thus 

^   dx3  =  third  differential  of  u. 
dx3 

Sign  of  the  first  differential  coefficient.  —  If  we  have  a  curve 
Artiose  equation  is  y  =  fx,  referred  to  rectangular  coordinates,  the  curve 

will  recede  from  the  axis  of  X  when   -~  is  positive,  and  approach  the 

axis  when  it  is  negative,  when  the  curve  lies  within  the  first  angle  of  the 
coordinate  axes.  For  all  angles  and  every  relation  of  y  and  x  the  curve 
will  recede  from  the  axis  of  X  when  the  ordinate  and  first  differential 
coefficient  have  the  same  sign,  and  approach  it  .when  they  have  different 
signs.  If  the  tangent  of  the  curve  becomes  parallel  to  the  axis  of  X  at  any 

point  ~-  =  0.    If  the  tangent  becomes  perpendicular  to  the  axis  of  X  at 

any  point  ^|  =  oo.  t 

Sign  of  the  second  differential  coefficient.  —  The  second  differential 
coefficient  has  the  same  sign  as  the  ordinate  when  the  curve  is  convex 
toward  the  axis  of  abscissa  and  a  contrary  sign  when  it  is  concave. 

Maclaurin's  Theorem.  —  For  developing  into  a  series  any  function 
of  a  single  variable  as  u  =  A  +  Bx  +  Cxz  +  Dx3  +  Ex*,  etc.,  in  which 
A,  B,  C,  etc.,  are  independent  of  x: 


In  applying  the  formula,  omit  the  expressions  x  «=  0,  although  the 
coefficients  are  always  found  under  this  hypothesis. 
EXAMPLES: 

(a  +  x)m  =  am  +  mam~l  x  + 


.  -  i  -  4  + 


a  +  x       a       a2       a3       a4  an  +  1 

Taylor's  Theorem.  —  For  developing  into  a  series  any  function  of  the 
sum  or  difference  of  two  independent  variables,  as  u'  «=  j(x  ±  y): 


in  which  u  is  what  u'  becomes  when  y  —  0,  ~  is  what  •—•  becomes  when 

y  —  0,  etc. 

Maxima  and  minima.  —  To  find  the  maximum  or  minimum  value 
of  a  function  of  a  single  variable: 

1.  Find  the  first  differential  coefficient  of  the  function,  place  it  equal 
to  0,  and  determine  the  roots  of  the  equation. 

2.  Find  the  second  differential  coefficient,  and  substitute  each  real  root, 


DIFFERENTIAL  CALCULUS.  79 

In  succession,  for  the  variable  in  the  second  member  of  the  equation. 
Each  root  which  gives  a  negative  result  will  correspond  to  a  maximum 
value  of  the  function,  and  each  which  gives  a  positive  result  will  corre- 
spond to  a  minimum  value. 

EXAMPLE.  —  To  find  the  value  of  x  which  will  render  the  function  y  a 
maximum  or  minimum  in  the  equation  of  the  circle,  y2  +  x2  =  R2; 

f|  -  -  y;  making  -  jj  ~  0  gives  x  -  0. 

The  second  differential  coefficient  is:  ~  =  -  x  +3 y*  • 

When  x  —  0,  y  —  R;  hence    -^  =  -  ^»  which  being  negative,  y  is  a 

maximum  for  R  positive. 

In  applying  the  rule  to  practical  examples  we  first  find  an  expression  for 
the  function  which  is  to  be  made  a  maximum  or  minimum. 

2.  If  in  such  expression  a  constant  quantity  is  found  as  a  factor,  it  may 
be  omitted  in  the  operation;  for  the  product  will  be  a  maximum  or  a  mini- 
mum when  the  variable  factor  is  a  maximum  or  a  minimum. 

3.  Any  value  9f  the  independent  variable  which  renders  a  function  a 
maximum  or  a  minimum  will  render  any  power  or  root  of  that  function  a 
maximum  or  minimum;  hence  we  may  square  both  members  of  an  equa- 
tion to  free  it  of  radicals  before  differentiating. 

By  these  rules  we  may  find : 

The  maximum  rectangle  which  can  be  inscribed  in  a  triangle  is  one 
whose  altitude  is  half  the  altitude  of  the  triangle. 

The  altitude  of  the  maximum  cylinder  which  can  be  inscribed  in  a  cone 
is  one  third  the  altitude  of  the  cone. 

'The  surface  of  a  cylindrical  vessel  of  a  given  volume,  open  at  the  top, 
is  a  minimum  when  the  altitude  equals  half  the  diameter. 

The  altitude  of  a  cylinder  inscribed  in  a  sphere  when  its  convex  surface  is 
a  maximum  is  r  v^2.  r  =  radius. 

The  altitude  of  a  cylinder  inscribed  in  a  sphere  when  the  volume  is  a 
maximum  is  2r  •*•  Vs. 

Maxima  and  Minima  without  the  Calculus.  —  In  the  equation 
y  =  a :  4-  bx  +  ex2,  in  which  a,  &,  and  c  are  constants,  either  positive  or 
negative,  if  c  be  positive  y  is  a  minimum  when  x  =  —  b  -*-  2c;  if  c  be 
negative  y  is  a  maximum  when  x  =  —  b  •*•  2c.  In  the  equation  y  =  a  + 
bx  +c/x,  y  is  a,  minimum  when  bx  =  c/x. 

APPLICATION.  — The  cost  of  electrical  transmission  is  made  up  (1)  of 
fixed  charges,  such  as  superintendence,  repairs,  cost  of  poles,  etc.,  which 
may  be  represented  by  a;  (2)  of  interest  on  cost  of  the  wire,  which  varies 
with  the  sectional  area,  and  may  be  represented  by  bx;  and  (3)  of  cost  of 
the  energy  wasted  in  transmission,  which  varies  inversely  with  the  area 
of  the  wire,  or  c/x.  The  total  cost,  y  =  a  4-  bx  +  c/x,  is  a  minimum 
when  item  2  =  item  3,  or  bx  =  c/x. 

Differential  of  an  exponential  function. 

If  u  =  ax    .     .     . (1) 

then  du  =  dax  =  axkdx (2) 

in  which  k  is  a  constant  dependent  on  a. 

The  relation  bet  ween  a  and  k  is  o*  =  e;  whence  a  =  e*  ....  (3) 
in  which  e  =  2.7182818  .  .  .  the  base  of  the  Naperian  system  of  loga- 
rithms. 

Logarithms.  —  The  logarithms  in  the  Naperian  system  are  denoted  by 
I,  Nap.  log  or  hyperbolic  log,  hyp.  log,  or  loge  ;  and  in  the  common  system 
Iways  by  log. 

fc  —  Nap.  logo;  log  a     =  k  log  e (4) 


80  DIFFERENTIAL  CALCULUS. 

The  common  logarithm  of  e,  =  log  2.7182818  .  .  .  «*  0.4342945  .  .  .  ; 
Is  called  the  modulus  of  the  common  system,  and  is  denoted  by  Af. 
Hence,  if  we  have  the  Naperian  logarithm  of  a  number  we  can  find  the 
common  logarithm  of  the  same  number  by  multiplying  by  the  modulus. 
Reciprocally,  Nap.  log  =  com.  log  X  2.3025851. 

If  in  equation  (4)  we  make  a  =  10,  we  have 


1  =  k  log  e,  or  ?  =  log  e  =  M  . 


That  is,  the  modulus  of  the  common  system  is  equal  to  1,  divided  by  the 
Naperian  logarithm  of  the  common  base. 
From  equation  (2)  we  have 

du       dax 
—  *—  •-  kdx. 
u        ax 

If  we  make  a  =  10,  the  base  of  the  common  system,  x  =  log  u,  and 

j  /i  j        du  „    1       du  ^  f 

d  (log  u)  -  dx  =  ~  X  £  -  -  X  M. 

That  is,  the  differential  of  a  common  logarithm  of  a  quantity  is  equal  to 
the  differential  of  the  quantity  divided  by  the  quantity,  into  the  modulus. 
If  we  make  a  =  e,  the  base  of  the  Naperian  system,  x  becomes  the  Nape- 
rian logarithm  of  u,  and  k  becomes  1  (see  equation  (3));  hence  M  =  1, 
and 

du          du 

d  (Nap.  log  u)  =  dx  —  —  -  ;   =  —  • 
ax  u 

That  is,  the  differential  of  a  Naperian  logarithm  of  a  quantity  is  equal  to 
the  differential  of  the  quantity  divided  by  the  quantity;  and  in  the 
Naperian  system  the  modulus  is  1. 

Since  k  is  the  Naperian  logarithm  of  a,  du  =  ax  I  a  dx.  That  is,  the 
differential  of  a  function  of  the  form  ax  is  equal  to  the  function,  into  the 
Naperian  logarithm  of  the  base  .a,  into  the  differential  of  the  exponent. 

If  we  have  a  differential  in  a  fractional  form,  in  which  the  numerator  is 
the  differential  of  the  denominator,  the  integral  is  the  Naperian  logarithm 
of  the  denominator.  Integrals  of  fractional  differentials  of  other  forms 
are  given  below: 

Differential  forms  which  have  known  integrals;  exponential 
functions.  (I  =  Nap.  log.) 


+  C; 


4. 

6.  I  •  ,    "^         -  l(x  ±  a  +  Vx*~±  2ax)  +  C; 


CALCULUS. 


81. 


7. 


8. 


9. 


10. 


2a  cte 


-j 


f  —  |«^ 

J  x\/a*  +  x* 


2a<ta        =  l  la  -  Va2  -  x~A  +  C; 
a2  -  a:2         \  a  +  ^a2  -  z2/ 

r  *~2^    =_z  fi  +  vi  +  o«s«\  [  c 

x  -{-  x  »  * 


Circular  functions.  —  Let  z  denote  an  arc  in  the  first  quadrant,  y  its 
sine,  x  its  cosine,  y  its  versed  sine,  and  t  its  tangent;  and  the  following  nota- 
tion be  employed  to  designate  an  arc  by  any  one  of  its  functions,  viz., 

sin"1  y  denotes  an  arc  of  which  y  is  the  sine, 
cos"1^  "  "  "  "  "  x  is  the  cosine, 
tan"^  "  "  "  "  "  Ms  the  tangent, 

(read  "arc  whose  sine  is  ?/,"  etc.),  —  we  have  the  following  differential 
forms  which  have  known  integrals  (r  =  radius): 


|  cos  zdz        •=  sin  z+  C\ 
I  —  sin  z  dz    •=  cos  z  +  C\ 


sin  z  dz  =  versin  z  +  (7; 


The  cycloid.  —  If  a  circle  be  rolled  along  a  straight  line,  any  point  of 
the  circumference,  as  P,  will  describe  a  curve  which  is  called  a  cycloid. 
The  circle  is  called  the  generating  circle,  and  P  the  generating  point. 


SLIDE 


The  transcendental  equation  of  the  cycloid  is 


x  =  versin  ~i v 

and  the  differential  equation  is  dx  = 


The  area  of  the  cycloid  is  equal  to  three  times  the 
area  of  the  generating  circle. 

The  surface  described  by  the  arc  of  a  cycloid  when 
revolved  about  its  base  is  equal  to  64  thirds  of  the 
generating  circle. 

The  volume  of  the  solid  generated  by  revolving 
a  cycloid  about  its  base  is  equal  to  five  eighths  of  the 
circumscribing  cylinder. 

Integral  calculus.  —  In  the  integral  calculus  we 
have  to  return  from  the  differential  to  the  function 
from  which  it  was  derived.  A  number  of  differential 
expressions  are  given  above,  each  of  which  has  a 
known  integral  corresponding  to  it,  which,  being 
differentiated,  will  produce  the  given  differential. 

In  all  classes  of  functions  any  differential  expression 
may  be  integrated  when  it  is  reduced  to  one  of  the 
known  forms;  and  the  operations  of  the  integral  cal- 
culus consist  mainly  in  making  such  transformations 
of  given  differential  expressions  as  shall  reduce  them 
to  equivalent  ones  whose  integrals  are  known. 

For  methods  of  making  these  transformations 
reference  must  be  made  to  the  text-books  on  differen- 
tial and  integral  calculus. 


THE  SLIDE  RULE. 

The  slide  rule  is  based  on  the  principles  that  the 
addition  of  logarithms  multiplies  the  numbers  which 
they  represent,  and  subtracting  logarithms  divides 
the  numbers.  By  its  use  the  operations  of  multiplica- 
tion, division,  the  finding  of  powers  and  the  extraction 
of  roots,  may  be  performed  rapidly  and  with  an  ap- 
proximation to  accuracy  which  is  sufficient  for  many 
purposes.  With  a  good  10-inch  Mannheim  rule  the 
results  obtained  are  usually  accurate  to  1/4  of  1  per 
cent.  Much  greater  accuracy  is  obtained  with  cylin- 
drical rules  like  the  Thacher. 

The  rule  (see  Fig.  73)  consists  of  a  fixed  and  a 
sliding  part  both  of  which  are  ruled  with  logarithmic 
scales;  that  is,  with  consecutive  divisions  spaced  not 
equally,  as  in  an  ordinary  scale,  but  in  proportion 
to  the  logarithms  of  a  series  of  numbers  from  1  to 
10.  By  moving  the  slide  to  the  right  or  left  the  loga- 
rithms are  added  or  subtracted,  and  multiplication 
or  division  of  the  numbers  thereby  effected.  The 
scales  on  the  fixed  part  of  the  rule  are  known  as  the 
A  and  D  scales,  and  those  on  the  slide  as  the  B  and 
C  scales.  A  and  B  are  the  upper  and  C  and  D 
are  the  lower  scales.  The  A  and  B  scales  are  each 
divided  into  two,  left  hand  and  right  hand,  each 
being  a  reproduction,  one  half  the  size,  of  the  C  and 
D  scales.  A  "runner,"  which  consists  of  a  framed 
glass  plate  with  a  fine  vertical  line  on  it,  is  used  to 
facilitate  some  of  the  operations.  The  numbering  on 
each  scale  begins  with  the  figure  1,  which  is  called 


FIG.  73. 


THE  SLIDE  RULE.  83 

the  "index"  of  the  scale.  In  using  the  scale  the  figures  1,  2,  3,  etc.,  are 
to  be  taken  either  as  representing  these  numbers,  or  as  10,  20,  30,  etc., 
100,  200,  300,  etc.,  0.1,  0.2,  0.3,  etc.,  that  is,  the  numbers  multiplied  or 
divided  by  10,  100,  etc.,  as  may  be  most  convenient  for  the  solution  of  a 
given  problem. 

The  following  examples  will  give  an  idea  of  the  method  of  using  the 
glide  rule,, 

Proportion.  —  Set  the  first  term  of  a  proportion  on  the  C  scale  opposite 
the  second  term  on  the  D  scale,  then  opposite  the  third  term  on  the  C 
scale  read  the  fourth  term  on  the  D  scale. 

EXAMPLE,  —  Find  the  fourth  term  in  the  proportion  12  :  21  ::  30  :  x. 
Move  the  slide  to  the  right  until  12  on  C  coincides  with  21  on  Z),  then 
opposite  30  on  C  read  x  on  D  =  52.5.  The  A  and  B  scales  may  be  used 
instead  of  C  and  D. 

Multiplication.  —  Set  the  index  or  figure  1  of  the  C  scale  to  one  of  the 
factors  on  ZX 

EXAMPLE.  —  25  X  3.  Move  the  slide  to  the  right  until  the  left  index 
of  C  coincides  with  25  on  the  D  scale.  Under  3  on  the  C  scale  will  be 
found  the  product  on  the  Z)  scale,  =  75. 

Division,  —  Place  the  divisor  on  C  opposite  the  dividend  on  D,  and  the 
quotient  will  be  found  on  D  under  the  index  of  C. 

EXAMPLE.  —  750  •*-  25.  Move  the  slide  to  the  right  until  25  on  C  coin- 
cides with  750  on  D.  Under  the  left  index  of  C  is  found  the  quotient  on 
D,  =  30. 

Combined  Multiplication  and  Division.  —  Arrange  the  factors  to  be 
multiplied  and  divided  in  the  form  of  a  fraction  with  one  more  factor  in 
the  numerator  than  in  the  denominator,  supplying  the  factor  1  if  necessary. 
Then  perform  alternate  division  and  multiplication,  using  the  runner  to 
Indicate  the  several  partial  results. 

4X5X8 
EXAMPLE^  —  —  3  ..  g     =  8.9  nearly.     Set   3  on  C  over  4  on  D,  set 

O  X   O 

runner  to  5  on  C,  then  set  6  on  C  under  the  runner,  and  read  under  8  on 
C  the  result  8*9  -  on  D. 

Involution  and  Evolution.  —  The  numbers  on  scales  A  and  B  are  the 
squares  of  their  coinciding  numbers  on  the  scales  C  and  D,  and  also  the 
numbers  on  scales  C  and  D  are  the  square  roots  of  their  coinciding  num- 
bers on  scales  A  and  B. 

EXAMPLE,  —  42  =  16.  Set  the  runner  over  4  on  scale  D  and  read  16 
on  A.__ 

^16  =  4.     Set  the  runner  over  16  on  A  and  read  4  on  D. 

In  extracting  square  roots,  if  the  number  of  digits  is  odd,  take  the  num- 
ber on  the  left-hand  scale  of  A ;  if  the  number  of  digits  is  even,  take  the 
number  on  the  right-hand  scale  of  A. 

To  cube  a  number  perform  the  operations  of  squaring  and  multiplica- 
tion. 

EXAMPLEO  —  2s  =  8.  Set  the  index  of  C  over  2  on  D,  and  above  2 
on  B  read  the  result  8  on  A0 

Extraction  of  the  Cube  Root.  —  Set  the  runner  over  the  number  on  A, 
then  move  the  slide  until  there  is  found  under  the  runner  on  B,  the  same 
number  which  is  found  under  the  index  of  C  on  D;  this  number  is  the 
cube  root  desired. 

EXAMPLE  —  ^8=2.  Set  the  runner  over  8  on  A,  move  the  slide 
along  until  the  same  number  appears  under  the  runner  on  B  and  under 
the  index  of  C  on  D;  this  will  be  the  number  2. 

Trigonometrical  Computations.  —  On  the  under  side  of  the  slide  (which 
is  reversible)  are  placed  three  scales,  a  scale  of  natural  sines  marked  St 
a  scale  of  natural  tangents  marked  T,  and  between  these  a  scale  of  equal 
parts.  To  use  these  scales,  reverse  the  slide,  bringing  its  under  side  to 
the  top.  Coinciding  with  an  angle  on  S  its  sine  will  be  found  on  At  and 
coinciding  with  an  angle  on  T  will  be  found  the  tangent  on  D.  Sines  and 
tangents  can  be  multiplied  or  divided  like  numbers. 


84  LOGARITHMIC   RULED    PAPER* 

LOGARITHMIC  RULED  PAPER. 

W.  F.  Durand   (Eng.  News,  Sept.  28,  1893.) 

As  plotted  on  ordinary  cross-section  paper  the  lines  which  express 
relations  between  two  variables  are  usually  curved,  and  must  be  plotted 
point  by  point  from  a  table  previously  computed.  It  is  only  where  the 
exponents  involved  in  the  relationship  are  unity  that  the  line  becomes 
straight  and  may  be  drawn  immediately  on  the  determination  of  two  of 
its  points.  It  is  the  peculiar  property  of  logarithmic  section  paper  that 
for  all  relationships  which  involve  multiplication,  division,  raising  to 
powers,  or  extraction  of  roots,  the  lines  representing  them  are  straight. 
Any  such  relationship  may  be  represented  by  an  equation  of  the  form: 
y  =  Bxn.  Taking  logarithms  we  have:  log  y  =  log  B  4-  n  log  x. 

Logarithmic  section  paper  is  a  short  and  ready  means  of  plotting  such 
logarithmic  equations.  The  scales  on  each  side  are  logarithmic  instead 
of  uniform,  as  in  ordinary  cross-section  paper.  The  numbers  and  divi- 
sions marked  are  placed  at  such  points  that  their  distances  from  the  origin 
are  proportional  to  the  logarithms  of  such  numbers  instead  of  to  the 
numbers  themselves.  If  we  take  any  point,  as  3,  for  example,  on  such  a 
scale,  the  real  distance  we  are  dealing  with  is  log  3  to  some  particular 
base,  and  not  3  itself.  The  number  at  the  origin  9f  such  a  scale  is  always 
1  and  not  0,  because  1  is  the  number  whose  logarithm  is  0.  This  1  may, 
however,  represent  a  unit  of  any  order,  so  that  quantities  of  any  size 
whatever  may  be  dealt  with. 

If  we  have  a  series  of  values  of  x  and  of  Bx  ,  and  plot  on  logarithmic 
section  paper  x  horizontally  and  Bxn  vertically,  the  actual  distances 
Involved  will  be  log  x  and  log  (Bxn),  or  log  B  +  n  log  x.  But  these  dis- 
tances will  give  a  straight  line  as  the  locus.  Hence  all  relationships 
expressible  in  this  form  are  represented  on  logarithmic  section  paper  by 
straight  lines.  It  follows  that  the  entire  locus  may  be  determined  from 
any  two  points;  that  is,  from  any  two  values  of  Bxn\  or,  again,  by  any  one 
point  and  the  angle  of  inclination;  that  is,  by  one  value  of  Bxn  and  the 
value  of  ft,  remembering  that  n  is  the  tangent  of  the  angle  of  inclination 
to  the  horizontal. 

A  single  square  plotted  on  each  edge  with  a  logarithmic  scale  from  1 
fco  10  may  be  made  to  serve  for  any  number  whatever  from  0  to  oo.  Thus 
to  express  graphically  the  locus  of  the  equation:  y  =  rrs/2.  Let  Fig.  74 
denote  a  square  cross-sectioned  with  logarithmic  scales,  as  described. 
Suppose  that  there  were  joined  to  it  and  to  each  other  on  the  right  and 
above,  an  indefinite  series  of  such  squares  similarly  divided.  Then,  con- 
sidering, in  passing  from  one  square  to  an  adjacent  one  to  the  right  or 
above,  that  the  unit  becomes  of  next  higher  order,  such  a  series  of  squares 
would,  with  the  proper  variation  of  the  unit,  represent  all  values  of  either 
x  or  y  between  0  and  oo, 

Suppose  the  original  square  divided  on  the  horizontal  edge  into  3  parts, 
and  on  the  vertical  edge  into  2  parts,  the  points  of  division  being  at  A, 
B,  D,  F,  G,  I.  Then  lines  joining  these  points,  as  shown,  will  be  at  an 
inclination  to  the  horizontal  whose  tangent  is  3/2.  Now,  beginning  at  0, 
OF  will  give  the  value  of  a^/2  for  values  of  x  from  1  to  that  denoted  by  HF, 
or  OB,  or  about  4.6.  For  greater  values  of  x  the  line  would  run  into  the 
adjacent  square  above,  but  the  location  of  this  line,  if  continued,  would 
be  exactly  similar  to  that  of  BD  in  the  square  before  us.  Therefore  the 
line  BD  will  give  values  of  :r3/2  for  x  between  B  and  C,  or  4.6  and  10,  the 
corresponding  values  of  y  being  of  the  order  of  tens,  and  ranging  from  10 
to  31.3.  For  larger  values  of  x  the  unit  of  x  is  of  the  higher  order,  and 
we  run  into  an  adjacent  square  to  the  right  without  change  of  unit  for  y. 
In  this  square  we  should  traverse  a  line  similar  to  IG.  Therefore,  by  a 
proper  choice  of  units  we  may  make  use  of  IG  for  the  determination  of 
values  of  £3/2  where  x  lies  between  10  and  the  value  at  G,  or  about  21.5. 
We  should  then  run  into  an  adjacent  square  above,  requiring  the  unit  on 
y  to  be  of  the  next  higher  order,  and  traverse  a  line  similar  to  AEt  which 


takes  us  finall 


LOGARITHMIC   RULED  PAPER. 


85 


takes  us  finally  to  the  opposite  corner  and  completes  the  cycle.  Follow- 
ing this,  the  same  series  of  lines  would  result  for  numbers  of  succeeding 
orders. 

The  value  of  x3/2  for  any  value  of  x  between  1  and  oo  may  thus  be  read 
from  one  or  another  of  these  lines,  and  likewise  for  any  value  between 
0  and  1.  The  location  of  the  decimal  point  is  readily  found  by  a  little 
attention  to  the  numbers  involved.  The  limiting  values  of  x  for  any 
given  line  may  be  marked  on  it,  thus  enabling  a  proper  choice  to  be  readily 
made.  Thus,  in  Fig.  74  we  mark  OF  as  0  -  4.6,  BD  as  4.6  -  10,  1G  as 


O      p 


10  —  21.5,  and  A E  as  21.5  —  100.  If  values  of  x  less  than  1  are  to  be 
dealt  with,  AE  will  serve  for  values  of  x  between  1  and  0.215,  IG  for 
values  between  0.215  and  0.1,  BD  for  values  between  0.1  and  0.046,  and 
OF  for  values  between  0.046  and  0.001. 

The  principles  involved  in  this  case  may  be  readily  extended  to  any 
other,  and  in  general  if  the  exponent  be  represented  by  m/n,  the  complete 
set  of  lines  may  be  drawn  by  dividing  one  side  of  the  square  into  m  and 
the  other  into  n  parts,  and  joining  the  points  of  division  as  in  Fig.  74.  In 
all  there  will  be  (m  -f  n  —  1)  lines,  and  opposite  t9  any  point  on  X  there 
will  be  n  lines  corresponding  to  the  n  different  beginnings  of  the  nth  root 


86  MATHEMATICAL  TABLES. 

of  the  mth  power,  while  opposite  to  any  point  on  Y  will  be  m  lines  corre- 
sponding to  the  different  beginnings  of  the  mth  root  of  the  nth  power. 
Where  the  complete  number  of  lines  would  be  quite  large,  it  is  usually 
unnecessary  to  draw  them  all,  and  the  number  may  be  limited  to  those 
necessary  to  cover  the  needed  range  in  the  values  of  x. 

If,  instead  of  the  equation  y  =  xnt  we  have  a  constant  term  as  a  multi- 
plier, giving  an  equation  in  the  more  general  form  y  =  Bxn,  or  Bx  m/n, 
there  will  be  the  same  number  of  lines  and  at  the  same  inclination,  but 
all  shifted  vertically  through  a  distance  equal  to  log  B.  If,  therefore, 
we  start  on  the  axis  of  Y  at  the  point  B,  we  may  draw  in  the  same  series 
of  lines  and  in  a  similar  manner.  In  this  way  PQ  represents  the  locus 
giving  the  values  of  the  areas  of  circles  in  terms  of  their  diameters,  being 
the  locus  of  the  equation  A  =  1/4  »  d2  or  y  =  1/4"'  £2- 

If  in  any  case  we  have  x  in  the  denominator  such  that  the  equation  is 
in  the  form  y  =  B/xn,  this  is  equal  to  y  =  Bx~n.  and  the  same  general 
rules  hold.  The  lines  in  such  case  slant  downward  to  the  right  instead  of 
upward.  Logarithmic  ruled  paper,  with  directions  for  the  use,  may  be 
obtained  from  Keuffel  &  Esser  Co.,  127  Fulton  St.,  New  York. 


MATHEMATICAL  TABLES. 

Formula  for  Interpolation. 

(n-1)  (n-2)   ^    ,    (n-1)  (n-2)  (n-3) 
—  -      --    -      —  ^     —         - 


(n  — 


— 


ai  =  the  first  term  of  the  series;  n,  number  of  the  required  term;  an,  the 
required  term;  di,  dz,  d3,  first  terms  of  successive  orders  of  differences 
between  ai,  a2,  a3,  a4,  successive  terms. 

EXAMPLE.  —  Required  the  log  of  40.7,  logs  of  40,  41  ,  42,  43  being  given  as 
below. 

Terms  alt  a2,  as,  a4(:     1.6021     1.6128     1.6232     1.6335 

1st  differences:  0.0107     0.0104     0.0103 
2d  -  0.0003  -  0.0001 

3d  +  0.0002 

For  log.  40,  n  =  1;  log  41,  n=  2;  for  log  40.7,  n  —  1.7;  n  —  1  =  0.7:  n  —  2 
=  -  0.3;  n  -  3  =-  1.3. 


an  =1.6021+0.7  (0.0107)  +(0.7)(-0.3K-0.0003)  +(0.7)(-0.3)(-  1.3)(0.0002!| 
=  1.6021  4-  0.00749  +  0.000031  4-  0.000009  =  1.6096  +. 


RECIPROCALS   OF  NUMBERS. 


RECIPROCALS  OF  NUMBERS. 


87 


No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

1 

1.00000000 

64 

01562500 

127 

•00787402 

190 

.00526316 

253 

.00395257 

2 

.50000000 

5 

01538461 

8 

•00781250 

1 

.C0523560 

4 

.00393701 

3 

.33333333 

6 

01515151 

9 

•00775194 

2 

.00520833 

5 

.00392157 

4 

.25000000 

7 

01492537 

130 

00769231 

3 

.00518135 

6 

.00390625 

5 

.20000000 

8 

01470588 

1 

00763359 

4 

.00515464 

7 

.00389105 

6 

.16666667 

9 

01449275 

2 

00757576 

5 

.00512820 

8 

.0038/597 

7 

.14285714 

70 

01428571 

3 

•00751880 

6 

.00510204 

9 

.00386100 

8 

.12500COO 

1 

•01408451 

4 

•00746269 

7 

.00507614 

260 

.00384615 

9 

11111111 

2 

•01388889 

5 

00740741 

8 

.00505051 

1 

.00383142 

10 

'.10000000 

3 

•01369863 

6 

•00735294 

9 

.00502513 

2 

.00381679 

11 

.09090909 

4 

•01351351 

7 

-00729927 

200 

.00500000 

3 

.00380228 

12 

.08333333 

5 

•01333333 

8 

-00724638 

1 

.00497512 

4 

.00378786 

13 

.07692308 

6 

01315789 

9 

-00719424 

2 

.00495049 

5 

.00377358 

14 

.07142857 

7 

•01298701 

140 

•00714286 

3 

.0049261  1 

6 

.00375940 

15 

.06666667 

8 

•01282051 

1 

•00709220 

4 

.004901% 

7 

.00374532 

16 

.06250000 

9 

•01265823 

2 

•00704225 

5 

.00487805 

8 

.00373134 

17 

.05882353 

80 

•01250000 

3 

-00699301 

6 

.00485437 

9 

.00371747 

18 

.05555556 

1 

•01234568 

4 

•00694444 

7 

.00483092 

270 

.00370370 

19 

.05263158 

2 

01219512 

5 

.00689655 

8 

.00480769 

1 

.00369004 

20 

.05000000 

3 

•01204819 

6 

.00684931 

9 

.00478469 

2 

.00367647 

.04761905 

4 

•01190476 

7 

.00680272 

210 

.00476190 

3 

.00366300 

^ 

.04545455 

£ 

01176471 

8 

.00675676 

11 

.00473934 

4 

.00364%3 

2 

.04347826 

e 

•01  162791 

9 

-00671141 

12 

.00471698 

5 

.00363636 

4 

.04166667 

7 

•01149425 

150 

-00666667 

13 

.00469484 

6 

.00362319 

c 

.04000000 

8 

•01136364 

1 

.00662252 

14 

.00467290 

7 

.00361011 

6 

.03846154 

9 

01  123595 

2 

.00657895 

15 

.00465116 

8 

.00359712 

T 

.03703704 

90 

•01111111 

3 

.00653595 

16 

.00462963 

9 

.00358423 

8 

.03571429 

1 

01098901 

4 

.00649351 

17 

.00460829 

280 

.00357143 

g 

.03448276 

2 

010S6956 

5 

.00645161 

18 

.00458716 

1 

.00355872 

30 

.03333333 

3 

•01075269 

6 

.00641026 

19 

.00456621 

2 

.00354610 

.03225806 

4 

•01063830 

7 

.00636943 

220 

.00454545 

3 

.00353357 

j 

.03125000 

c 

01052632 

8 

.0063291  1 

| 

.00452489 

4 

.00352113 

• 

.03030303 

6 

•01041667 

9 

.00628931 

2 

.00450450 

5 

.00350877 

4 

.02941176 

7 

•01030928 

160 

.00625000 

3 

.00448430 

6 

.0034%50 

« 

.02857143 

8 

•01020408 

1 

.00621118 

4 

.00446429 

7 

.00348432 

6 

.027/7778 

9 

•01010101 

2 

.00617284 

5 

.00444444 

8 

.00347222 

j 

.02702703 

100 

•01000000 

3 

.00613497 

6 

.00442478 

9 

.00346021 

8 

.02631579 

1 

•00990099 

4 

.00609756 

7 

.00440529 

290 

.00344828 

g 

.02564103 

2 

•00980392 

5 

.00606061 

8 

.004385% 

1 

.00343643 

40 

.02500000 

2 

•00970874 

6 

.00602410 

9 

.00436681 

2 

.00342466 

| 

.02439024 

A 

•00%  1538 

7 

.00598802 

230 

.00434783 

3 

.00341297 

'4 

.02380952 

c 

•00952381 

8 

.00595238 

1 

.00432900 

4 

.00340136 

\ 

.02325581 

t 

•009433% 

9 

00591716 

2 

.00431034 

5 

.00338983 

t 

.02272727 

7 

.00934579 

170 

.00588235 

3 

.00429184 

6 

.00337838 

i 

02222222 

8 

-00925926 

1 

.00584795 

4 

.00427350 

7 

.00336700 

6 

.02173913 

9 

.00917431 

2 

.00581395 

5 

.00425532 

8 

.00335570 

7 

.02127660 

110 

.00909091 

3 

.00578035 

6 

.00423729 

9 

00334448 

8 

.02083333 

11 

.00900901 

4 

.00574713 

7 

.00421941 

300 

.00333333 

c 

.02040816 

12 

.00892857 

5 

.00571429 

8 

.00420168 

.00332226 

50 

.02000000 

13 

.00884956 

6 

.00568182 

9 

.00418410 

2 

.00331126 

1 

.01960784 

14 

.00877193 

7 

.00564972 

240 

.00416667 

3 

.00330033 

j 

.01923077 

15 

.00869565 

8 

.00561798 

1 

.00414938 

4 

.00328947 

- 

.01886792 

16 

.00862069 

9 

.00558659 

2 

.00413223 

5 

.00327869 

/ 

.01851852 

17 

.00854701 

180 

.00555556 

3 

.00411523 

6 

.00326797 

c 

.01818182 

18 

.00847458 

1 

.00552486 

4 

.00409836 

7 

.00325733 

( 

.01785714 

19 

.00840336 

2 

.00549451 

5 

.00408163 

8 

.00324675 

7 

.01754386 

120 

.00833333 

3 

.00546448 

6 

.00406504 

9 

.00323625 

8 

.01724138 

1 

.00826446 

4 

.00543478 

7 

.00404858 

310 

.00322581 

9 

.01694915 

2 

.00819672 

5 

.00540540 

8 

.00403226 

11 

.00321543 

60 

.01666667 

3 

.00813008 

6 

.00537634 

9 

.00401606 

12 

.00320513 

1 

.01639344 

4 

.00806452 

7 

.00534759 

250 

.00400000 

-13 

.00319489 

2 

.01612903 

5 

.00800000 

8 

.00531914 

1 

.00398406 

14 

.00318471 

3 

.01587302 

6 

.00793651 

9 

.00529100 

Z 

.00396825 

15 

.00317460 

88 


MATHEMATICAL  TABLES. 


No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

~  316 

.00316456 

381 

.00262467 

446 

.00224215 

511 

.00195695 

576 

.00173611 

17 

.00315457 

2 

.00261780 

7 

.00223714 

12 

.00195312 

7 

.00173310 

18 

.00314465 

3 

.00261097 

8 

.00223214 

13 

.00194932 

8 

.00173010 

19 

.00313480 

4 

.00260417 

9 

.00222717 

14 

.00194552 

9 

.00172712 

320 

.00312500 

5 

.00259740 

450 

.00222222 

15 

.00194175 

580 

.00172414 

1 

.00311526 

6 

.00259067 

1 

.00221729 

16 

.00193798 

1 

.00172117 

2 

.00310559 

7 

.00258398 

2 

.00221239 

17 

.00193424 

2 

.00171821 

3 

.00309597 

8 

.00257732 

3 

.00220751 

18 

.00193050 

3 

.00171527 

4 

.00308642 

9 

.00257069 

4 

.00220264 

19 

.00192678 

4 

.00171233 

5 

.00307692 

390 

.00256410 

5 

.00219780 

520 

.00192308 

5 

.00170940 

6 

.00306748 

1 

.00255754 

6 

.00219298 

1 

00191939 

6 

.00170648 

7 

.00305810 

2 

.00255102 

7 

.00218818 

2 

.00191571 

7 

.00170358 

8 

.00304878 

3 

.00254453 

8 

.00218341 

3 

.00191205 

8 

.00170068 

9 

.00303951 

4 

.00253807 

9 

.00217865 

4 

00190840 

9 

.00169779 

330 

.00303030 

5 

.00253165 

460 

.00217391 

5 

.00190476 

590 

.00169491 

1 

.00302115 

6 

.00252525 

1 

.00216920 

6 

.00190114 

1 

.00169205 

2 

.00301205 

7 

.00251889 

2 

.00216450 

7 

.00189753 

2 

.00168919 

3 

.00300300 

8 

.00251256 

3 

.00215983 

8 

.00189394 

3 

.00168634 

4 

.00299401 

9 

.00250627 

4 

.00215517 

9 

.00189036 

4 

.00168350 

5 

.00298507 

400 

.00250000 

5 

.00215054 

530 

.00188679 

5 

.00168067 

6 

.00297619 

1 

.00249377 

6 

.00214592 

1 

.00188324 

6 

.00167785 

7 

.00296736 

2 

.00248756 

7 

.00214133 

2 

.00187970 

7 

.00167504 

8 

.00295858 

3 

.00248139 

8 

.00213675 

3 

.00187617 

8 

.00167224 

9 

.00294985 

4 

.00247525 

9 

.00213220 

4 

.00187266 

9 

.00166945 

340 

.00294118 

5 

.00246914 

470 

.00212766 

5 

.00186916 

600 

.00166667 

1 

.00293255 

6 

.00246305 

1 

.00212314 

6 

.00186567 

1 

00166389 

2 

.00292398 

7 

.00245700 

2 

.00211864 

7 

.00186220 

2 

.00166113 

3 

.00291545 

8 

.00245098 

3 

.00211416 

8 

.00185874 

3 

.00165837 

4 

.00290698 

9 

.00244499 

4 

.00210970 

9 

.00185528 

4 

.00165563 

5 

.00289855 

410 

.00243902 

5 

.00210526 

540 

.00185185 

5 

.00165289 

6 

.00289017 

11 

.00243309 

6 

.00210084 

1 

.00184843 

6 

00165016 

7 

.00288184 

12 

.00242718 

7 

.00209644 

2 

.00184502 

7 

.00164745 

8 

.00287356 

13 

.00242131 

8 

.00209205 

3 

.00184162 

8 

.00164474 

9 

.00286533 

14 

.00241546 

9 

.00208768 

4 

.00183823 

9 

.00164204 

350 

,00285714 

15 

.00240964 

480 

.00208333 

5 

.00183486 

610 

.00163934 

1 

.00284900 

16 

.00240385 

1 

.00207900 

6 

00183150 

11 

00163666 

2 

.00284091 

17 

.00239808 

2 

.00207469 

7 

.00182815 

12 

.00163399 

3 

.00283286 

18 

.00239234 

3 

.00207039 

8 

.00182482 

13 

00163132 

4 

.00282486 

19 

.00238663 

4 

.00206612 

9 

00182149 

14 

.00162866 

5 

.00281690 

420 

.00238095 

5 

.00206186 

550 

.00181818 

15 

.00162602 

6 

.00280899 

1 

.00237530 

6 

.00205761 

1 

00181488 

16 

00162338 

7 

.00280112 

2 

.00236967 

7 

.00205339 

2 

.00181159 

17 

.00162075 

8 

.00279330 

3 

.00236407 

8 

.00204918 

3 

.00180832 

18 

00161812 

9 

.00278551 

4 

.00235849 

9 

.00204499 

4 

00180505 

19 

.00161551 

360 

.00277778 

5 

.00235294 

490 

.00204082 

5 

.00180180 

620 

00161290 

1 

.00277008 

6 

.00234742 

1 

.00203666 

6 

00179856 

1 

.00161031 

2 

.00276243 

7 

.00234192 

2 

.00203252 

7 

.00179533 

2 

00160772 

3 

.00275482 

8 

.00233645 

3  .00202840 

8 

00179211 

.  3 

00160514 

4 

.00274725 

9 

.00233100 

4 

.00202429 

9 

.00178891 

4 

.00160256 

5 

.00273973 

430 

.00232558 

5 

.00202020 

560 

.00178571 

5 

00160000 

6 

.00273224 

1 

.00232019 

6 

.00201613 

1 

.00178253 

6 

.00159744 

7 

.00272480 

2 

.00231481 

7 

.00201207 

2 

.00177936 

7 

00159490 

8 

.00271739 

3 

.00230947 

8 

.00200803 

3 

.00177620 

8 

.00159236 

9 

.00271003 

4 

.00230415 

9 

.00200401 

4 

.00177305 

9 

00158982 

370 

.00270270 

5 

.00229885 

500 

.00200000 

5 

.00176991 

630 

.00158730 

1 

.00269542 

6 

.00229358 

1 

.00199601 

6 

.00176678 

1 

00158479 

2 

.00268817 

7 

.00228833 

2 

.00199203 

7 

00176367 

2 

.00158228 

^ 

.00268096 

8 

.00228310 

3 

.00198807 

8 

.00176056 

3 

00157978 

4 

.00267380 

9 

.00227790 

4 

.00198413 

9 

.00175747 

4 

.00157729 

e 

.00266667 

440 

.00227273 

5 

.00198020 

570 

.00175439 

5 

00157480 

t 

.00265957 

1 

.00226757 

6 

.00197628 

1 

00175131 

6 

.00157233 

7 

.00265252 

2 

.00226244 

7 

.00197239 

2 

.00174825 

7 

.00156986 

8 

.00264550 

3 

.00225734 

8 

.00196850 

3 

.00174520 

8 

.00156740 

9 

.00263852 

4 

.00225225 

9 

.00196464 

.00174216 

9 

.00156494 

380 

.00263158 

5 

.00224719 

510 

.00196078 

5 

.00173913 

640 

.00156250 

RECIPROCALS   OF  NUMBERS. 


89 


No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipnx 
cal. 

641 

.001560% 

706 

.00141643 

771 

.00129702 

836 

.00119617 

901 

.001  10988 

2 

.00155763 

7 

00141443 

2 

.00129534 

7 

.001  19474 

2 

.001  10865 

3 

.00155521 

8 

00141243 

3 

.00129366 

8 

.00119332 

3 

.00110742 

4 

.00155279 

9 

.00141044 

4 

.00129199 

9 

.00119189 

4 

.00110619 

5 

.00155039 

710 

00140345 

5 

.00129032 

840 

.00119048 

5 

.00110497 

6 

.00154799 

11 

.00140647 

6 

.00128866 

1 

.001  18906 

6 

.001  10375 

7 

.00154559 

12 

.00140449 

7 

.00128700 

2 

.001  18765 

7 

.00110254 

8 

.00154321 

13 

.00140252 

8 

.00128535 

3 

.001  18624 

8 

.00110132 

9 

.00154033 

14 

.00140056 

9 

.00128370 

4 

.00118483 

9 

.00110011 

650 

.00153846 

15 

.00139860 

780 

.00128205 

5 

.001  18343 

910 

.00109890 

1 

.00153610 

16 

.00139665 

1 

.00128041 

6 

.001  18203 

11 

.00109769 

2 

.00153374 

17 

.00139470 

2 

.00127877 

7 

.00118064 

12 

.00109649 

3 

.00153140 

18 

.00139276 

3 

.00127714 

8 

.00117924 

13 

.00109529 

4 

.00152905 

19 

.00139032 

4 

.00127551 

9 

.00117786 

14 

.00109409 

5 

.00152672 

720 

.00138889 

5 

.00127388 

850 

.00117647 

15 

.00109290 

6 

.00152439 

1 

.00138696 

6 

.00127226 

1 

.00117509 

16 

.00109170 

7 

.00152207 

2 

.00138504 

7 

.00127065 

2 

.00117371 

17 

.00109051 

8 

.00151975 

3 

.00138313 

8 

.00126904 

3 

.00117233 

18 

.00108932 

9 

.00151745 

4 

.00138121 

9 

.00126743 

4 

.001170% 

19 

00108814 

660 

.00151515 

5 

00137931 

790 

.00126582 

5 

.001  16959 

920 

.00108696 

.00151236 

6 

.00137741 

1 

.00126422 

6 

.001  16822 

1 

.00108578 

2 

.00151057 

7 

.00137552 

2 

.00126263 

7 

.001  16686 

2 

.00108460 

3 

.00150330 

8 

.00137363 

3 

.00126103 

8 

.00116550 

3 

.00108342 

4 

.00150602 

9 

.00137174 

4 

.00125945 

9 

.00116414 

4 

.00108225 

5 

.00150376 

730 

00136936 

5 

.00125786 

860 

.00116279 

5 

.00108108 

6 

.00150150 

1 

.00136799 

6 

.00125628 

1 

.00116144 

6 

.00107991 

7 

.00149925 

2 

00136612 

7 

.00125470 

2 

.00116009 

7 

.00107875 

8 

.00149701 

3 

.00136426 

8 

.00125313 

3 

.00115875 

8 

.00107759 

9 

.00149477 

4 

00136240 

9 

.00125156 

4 

.00115741 

9 

.00107.643 

670 

.00149254 

5 

.00136054 

800 

.00125000 

5 

.00115607 

930 

.00107527 

1 

.00149031 

6 

00135870 

1 

.00124844 

6 

.00115473 

1 

.00107411 

2 

.00148809 

7 

.00135685 

2 

.00124688 

7 

.00115340 

2 

.00107296 

4 

.00148588 

8 

00135501 

3 

.00124533 

8 

.00115207 

3 

.00107181 

4 

.00148368 

9 

.00135318 

4 

.00124378 

9 

.00115075 

4 

.00107066 

j 

.00148148 

740 

00135135 

5 

.00124224 

870 

.001  14942 

5 

.00106952 

( 

.00147929 

1 

.00134953 

6 

.00124069 

1 

.00114811 

6 

.00106838 

7 

.00147710 

2 

00134771 

7 

.00123916 

2 

.00114679 

7 

.00106724 

8 

.00147493 

3 

.00134589 

8 

.00123762 

3 

.00114547 

8 

.00106610 

9 

.00147275 

4 

.00134409 

9 

.00123609 

4 

.00114416 

9 

.00106496 

680 

.00147059 

c 

.00134228 

810 

.00123457 

5 

.00114286 

940 

.00106383 

1 

.00146843 

6 

00134048 

11 

.00123305 

6 

.00114155 

1 

.00]  06270 

2 

.00146628 

7 

.00133869 

12 

.00123153 

7 

.00114025 

2 

.00106157 

3 

.00146413 

8 

00133690 

13 

.00123001 

8 

.00113895 

3 

00106044 

z 

.00146199 

9 

.00133511 

14 

.00122850 

9 

.00113766 

4 

.00105932 

c 

.00145985 

750 

00133333 

15 

.00122699 

880 

.00113636 

5 

00105820 

( 

.00145773 

1 

.00133156 

16 

.00122549 

1 

.00113507 

6 

00105708 

j 

.00145560 

2 

00132979 

17 

.00122399 

2 

.00113379 

7 

00105597 

8 

.00145349 

.00132802* 

18 

.00122249 

3 

.00113250 

8 

.00105485 

9 

.00145137 

4 

00132626 

19 

.00122100 

4 

.00113122 

9 

00105374 

690 

.00144927 

c 

.00132450 

820 

.00121951 

5 

.001  12994 

950 

.00105263 

J 

.00144718 

6 

00132275 

] 

.00121803 

6 

.00  H  2867 

1 

.00105152 

4 

.00144509 

7 

.00132100 

2 

.00121654 

7 

.00112740 

2 

.00105042 

2 

.00144300 

8 

.00131926 

3 

.00121507 

8 

.00112613 

3 

.00104932 

4 

.00144092 

9 

.00131752 

4 

.00121359 

9 

.001  12486 

4 

.00104822 

c 

.00143885 

760 

.00131579 

5 

.00121212 

890 

.001  12360 

5 

.00104712 

6 

.00143678 

1 

.00131406 

6 

.00121065 

1 

.00112233 

6 

00104602 

7 

.00143472 

2 

.00131234 

7 

.00120919 

2 

.00112108 

7 

.00104493 

8 

.00143266 

3 

.00131062 

8 

.00120773 

3 

.001  1  1982 

8 

00104384 

9 

.00143061 

4 

.00130890 

9 

.00120627 

4 

00111857 

9 

.00104275 

700 

.00142857 

5 

.00130719 

830 

.00120482 

5 

.00111732 

960 

.00104167 

1 

.00142653 

6 

.00130548 

1 

.00120337 

6 

.00111607 

1 

.00104058 

2 

.00142450 

7 

00130378 

2 

.00120192 

7 

.001  1  1483 

2 

.00103950 

3 

.00142247 

8 

.00130208 

3 

.00120048 

8 

.00111359 

3 

.00103842 

4 

.00142045 

9 

.00130039 

4 

.00119904 

9 

.00111235 

4 

.00103734 

3 

.00141844 

770 

.00129870 

5 

.001  19760 

900 

.00111111 

5 

.00103627 

00 


MATHEMATICAL  TABLES. 


No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 

No. 

Recipro- 
cal. 

.000815661 

966 

.00103520 

1031 

.000969932 

10% 

.000912409 

1161 

.000861326 

1226 

7 

.00103413 

2 

.000968992 

7 

.000911577 

2 

.000860585 

7 

.0008149% 

8 

.00103306 

3 

.000968054 

8 

.000910747 

3 

.000859845 

8 

.000814332 

9 

.00103199 

4 

.000967118 

9 

.000909918 

4 

.000859106 

9 

.000813670 

970 

.00103093 

5 

.000966184 

1100 

.000909091 

5 

.000858369 

1230 

.000813008 

I 

.00102987 

6 

.000965251 

1 

.000908265 

6 

.000857633 

1 

.000812348 

2 

.00102881 

7 

.000964320 

2 

000907441 

7 

.000856898 

2 

.000811688 

3 

.00102775 

8 

.000963391 

3 

.000906618 

8 

.000856164 

3 

.000811030 

4 

.00102669 

9 

.000962464 

4 

.000905797 

9 

.000855432 

4 

.000810373 

5 

.00102564 

1040 

000%  1538 

5 

.000904977 

1170 

.000854701 

5 

.000809717 

6 

.00102459 

1 

.000960615 

6 

.000904159 

1 

.000853971 

6 

.000809061 

7 

.00102354 

2 

.000959693 

7 

.000903342 

2 

.000853242 

7 

.000808407 

8 

.00102250 

3 

.000958774 

8 

.000902527 

3 

.000852515 

8 

.000807754 

9 

.00102145 

4 

.000957854 

9 

.000901713 

4 

.000851789 

9 

.000807102 

980 

.00102041 

5 

.000956938 

1110 

.000900901 

5 

.000851064 

1240 

.000806452 

1 

.00101937 

6 

.000956023 

1  1 

.000900090 

6 

.000850340 

1 

.000805802 

2 

.00101833 

7 

.000955110 

12 

.000899281 

7 

.00084%  18 

2 

.000805153 

3 

.00101729 

8 

.000954198 

13 

.000898473 

8 

.0008488% 

3 

.000804505 

4 

.00101626 

9 

.000953289 

14 

.000897666 

9 

.000848176 

4 

.000803858 

5 

.00101523 

1050 

.000952381 

15 

0008%861 

1180 

.000847457 

5 

.000803213 

6 

.00101420 

1 

.000951475 

16 

.0008%057 

1 

.000846740 

6 

.000802568 

7 

.00101317 

2 

.000950570 

17 

.000895255 

2 

.000846024 

7 

.000801925 

8 

.00101215 

3 

.000949668 

18 

.000894454 

3 

000845308 

8 

.000801282 

9 

.00101112 

4 

.000948767 

191.000893655 

4 

.000844595 

9 

.000800640 

990 

.00101010 

5 

.000947867 

1120 

000892857 

5 

.000843882 

1250 

.000800000 

1 

.00100908 

6 

.000946970 

1 

.000892061 

6 

.000843170 

1 

.000799360 

2 

.00100806 

7 

.000946074 

2 

.000891266 

7 

.000842460 

2 

.000798722 

3 

.00100705 

8 

.000945180 

3 

.000890472 

8 

.000841751 

3 

.000798085 

4 

.00100604 

9 

.000944287 

4 

.00088%80 

9 

.000841043 

4 

.000797448 

5 

.00100502 

1060 

.0009433% 

5 

.000888889 

1190 

.000840336 

5 

.0007%813 

6 

.00100402 

1 

.000942507 

6 

.000888099 

1 

00083%31 

6 

.0007%  178 

7 

.00100301 

2 

.000941620 

7 

.000887311 

2 

.000838926 

7 

.000795545 

8 

.00100200 

3 

.000940734 

8 

.000886525 

3 

.000838222 

8 

.000794913 

9 

.00100100 

4 

.000939850 

9 

.000885740 

4 

.000837521 

9 

.000794281 

1000 

.00100000 

5 

.000938%7 

1130 

.000884956 

5 

.000836820 

1260 

.000793651 

1 

.000999001 

6 

000938086 

1 

.000884173 

6 

.000836120 

1 

.000793021 

2 

.000998004 

7 

.000937207 

2 

.000883392 

7 

.000835422 

2 

.000792393 

3 

.000997009 

8 

.000936330 

3 

.000882612 

8 

.000834724 

3 

.000791766 

4 

.000996016 

9 

.000935454 

4 

.000881834 

9 

.000834028 

4 

000791139 

5 

.000995025 

1070 

.000934579 

5 

.000881057 

1200 

.000833333 

5 

.000790514 

6. 

.000994036 

1 

.000933707 

6 

000880282 

] 

000832639 

6 

.000789889 

7 

.000993049 

2 

.000932836 

7 

.000879508 

2 

.000831947 

7 

.000789266 

8 

.000992063 

3 

.00093  1%6 

8 

.000878735 

3 

.000831255 

8 

.000788643 

9 

.000991080 

4 

000931099 

9 

.000877%3 

4 

.000830565 

9 

000788022 

1010 

.000990099 

5 

.000930233 

1140 

.000877193 

5 

.000829875 

1270 

.000787402 

11 

.000989120 

6 

.000929368 

1 

000876424 

6 

000829187 

1  000786782 

12 

.000988142 

7 

.000928505 

2 

.000875657 

7 

.000828500 

21.000786163 

13 

.000987167 

8 

.000927644 

3 

.000874891 

8 

.000827815 

3|.000785546 

14 

.000986193 

9 

000926784 

4 

.000874126 

9 

.000827130 

4 

000784929 

15 

.000985222 

1080 

.000925926 

5 

.000873362 

1210 

.000826446 

5 

000784314 

16 

.000984252 

1 

000925069 

6 

000872600 

11  .000825764 

6 

.UUU/tf>699 

17 

.000983284 

2 

.000924214 

7 

.000871840 

12  .000825082 

7 

.000783085 

18 

.000982318 

3 

.000923361 

8 

.000871080 

13  000824402 

8 

000782473 

19 

.000981354 

4 

.000922509 

9 

.000870322 

14 

000823723 

9 

.000781861 

1020 

.000980392 

5 

.000921659 

1150 

.000869565 

15 

.000823045 

1280 

000781250 

1 

.000979432 

6 

000920810 

1 

000868810 

161.000822368 

1 

.000780640 

2 

.000978474 

7 

.0009  19%3 

2 

.000868056 

17 

.000821693 

2 

.000780031 

3 

.000977517 

8 

.000919118 

3 

.000867303 

18 

.000821018 

3 

.000779423 

4 

.000976562 

9 

.000918274 

4 

.000866551 

19 

.000820344 

4 

.000778816 

5 

.000975610 

1090 

.000917431 

5 

.000865801 

1220 

.0008  1%72 

5 

.000778210 

6 

.000974659 

1 

.000916590 

6 

.000865052 

1 

.000819001 

6 

.000777605 

7 

.000973710 

2 

.00091575 

7 

.000864304 

2 

.000818331 

7 

.000777001 

8 

.000972763 

3 

.000914913 

8 

.000863558 

3 

.000817661 

8 

.000776397 

9 

.000971817 

4 

.00091407: 

9 

.000862813 

4 

000816993 

9 

.000775795 

1030 

.000970874 

5 

.000913242 

1160 

.000862069 

5 

.000816326 

1290 

.000775194 

RECIPROCALS   OP  NUMBERS. 


91 


No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 

No. 

Recipro- 

No. 

Recipro* 
cal. 

1291 

.000774593 

1356 

.000737463 

1421 

.000703730 

i486 

.000672948 

1551 

.000644745 

2 

.000773994 

7 

.000736920 

2 

.000703235 

7 

.000672495 

2 

.000644330 

3 

.000773395 

8 

.000736377 

3 

.000702741 

8 

.000672043 

? 

.000643915 

4 

.000772797 

9 

.000735835 

4 

.000702247 

g 

.000671592 

t 

.000643501 

5 

.000772201 

1360 

.000735294 

5 

.000701754 

1490 

.000671141 

c 

.000643087 

6 

.000771605 

1 

.000734754 

6 

.000701262 

1 

.000670691 

\ 

.000642673 

7 

.000771010 

2 

.000734214 

7 

.000700771 

2 

.000670241 

7 

.000642261 

8 

.000770416 

3 

.000733676 

8 

.000700280 

3 

.000669792 

8 

.000641848 

9 

.000769823 

4 

.000733138 

9 

.000699790 

4 

.000669344 

9 

.000641437 

1300 

.000769231 

5 

.000732601 

1430 

.000699301 

5 

.0006688% 

1560 

.000641026 

1 

.000768639 

6 

.000732064 

1 

.000698812 

6 

.000668449 

.000640615 

2 

.000768049 

7 

.000731529 

2 

.000698324 

7 

.000668003 

2 

.000640205 

3 

.000767459 

8 

.000730994 

3 

.000697837 

8 

.000667557 

g 

.000639795 

4 

.000766871 

9 

.000730460 

4 

.000697350 

9 

.000667111 

^ 

.000639386 

5 

.000766283 

1370 

.000729927 

5 

.000696864 

1500 

.000666667 

• 

.000638978 

6 

.000765697 

1 

.000729395 

6 

000696379 

1 

.000666223 

t 

.000638570 

7 

.000765111 

2 

.000728863 

7 

000695894 

2 

.000665779 

7 

.000638162 

8 

.000764526 

3 

.000728332 

8 

000695410 

3 

.000665336 

8 

.000637755 

9 

.000763942 

4 

.000727802 

9 

000694927 

4 

.000664894 

9 

.000637349 

1310 

.000763359 

5 

.000727273 

1440 

000694444 

5 

.000664452 

1570 

.000636943 

11 

.000762776 

6 

.000726744 

1 

000693962 

6 

.000664011 

1 

.000636537 

12 

.000762195 

7 

.000726216 

2 

000693481 

7 

.000663570 

2 

.000636132 

13 

.000761615 

8 

.000725689 

3 

000693001 

8 

.000663130 

3 

.000635728 

14 

.000761035 

9 

.000725163 

4 

000692521 

9 

.000662691 

4 

.000635324 

15 

.000760456 

1380 

.000724638 

5 

000692041 

1510 

000662252 

5 

.000634921 

16 

.000759878 

1 

.000724113 

6 

000691563 

11 

000661813 

6 

.000634518 

17 

.000759301 

2 

.000723589 

7 

000691085 

12 

000661376 

7 

.000634115 

18 

.000758725 

3 

.000723066 

8 

000690608 

13 

000660939 

8 

.000633714 

19 

.000758150 

4 

.000722543 

9 

000690131 

14 

000660502 

9 

.000633312 

1320 

.000757576 

5 

.000722022 

1450 

000689655 

15 

000660066 

580 

.000632911 

.000757002 

6 

.000721501 

1 

000689180 

16 

00065%31 

.000632511 

2 

.000756430 

7 

.000720980 

2 

000688705 

17 

0006591% 

2 

.000632111 

3 

.000755858 

8 

.000720461 

3 

000688231 

18 

000658761 

3 

000631712 

4 

.000755287 

9 

.000719942 

4 

000687758 

19 

000658328 

4 

.000631313 

5 

.000754717 

1390 

.000719424 

5 

000687285 

1520 

000657895 

5 

000630915 

6 

.000754148 

1 

.000718907 

6 

000686813 

1 

000657462 

6 

000630517 

7 

.000753579 

2 

.000718391 

7 

000686341 

2 

000657030 

7 

C00630I20 

8 

.000753012 

3 

.000717875 

8 

000685871 

3 

000656598 

8 

000629723 

9 

.000752445 

4 

.000717360 

9l  000685401 

4 

000656168 

9 

000629327 

1330 

.000751880 

5 

.000716846 

1460 

.000684932 

5 

000655738 

590 

000628931 

1 

.000751315 

6 

000716332 

1 

000684463 

6 

000655308 

1 

000628536 

2 

.000750750 

7 

.000715820 

2 

.000683994 

7 

000654879 

2 

000628141 

3 

.000750187 

8 

000715308 

3 

.000683527 

8 

000654450 

3 

000627746 

4 

.000749625 

9 

.0007147% 

4 

.000683060 

9 

000654022 

4 

000627353 

5 

.000749064 

1400 

.000714286 

5 

000682594 

1530 

000653595 

5 

000626959 

6 

.000748503 

1 

.000713776 

6 

.000682128 

1 

000653168 

6 

000626566 

7 

.000747943 

2 

.000713267 

7 

.000681663 

2 

000652742 

7 

000626174 

8 

.000747384 

3 

.000712758 

8 

.000681199 

3 

000652316 

8 

000625782 

9 

.000746826 

4 

.000712251 

9 

.000680735 

4 

000651890 

9 

000625391 

1340 

.000746269 

5 

.000711744 

1470 

.000680272 

5 

000651466 

600 

000625000 

1 

.000745712 

6 

.000711238 

1 

.000679810 

6 

000651042 

2 

000624219 

2 

.000745156 

7 

.000710732 

2 

.000679348 

7 

000650618 

4 

000623441 

3 

.000744602 

8 

.000710227 

3 

000678887 

8 

000650195 

6 

000622665 

4 

.000744048 

9 

.000709723 

4 

.000678426 

9 

000649773 

8 

000621890 

5 

.000743494 

1410 

.000709220 

5 

000677966 

1540 

000649351 

610 

000621  1  18 

6 

.000742942 

11 

.000708717 

6 

.000677507 

1 

000648929 

12 

000620347 

7 

.000742390 

12 

.000708215 

7 

.000677048 

2 

000648508 

14 

000619578 

8 

.000741840 

13 

.000707714 

8 

.000676590 

3 

000648088 

16 

000618812 

9 

.000741290 

14 

.000707214 

9 

.000676132 

4 

000647668 

18 

000618047 

1350 

.000740741 

15 

.000706714 

1480 

.000675676 

5 

000647249 

620 

000617284 

1 

.000740192 

16 

.000706215 

1 

.000675219 

6 

000646830 

2 

000616523 

2 

.000739645 

17 

.000705716 

2 

.000674764 

7 

000646412 

A 

000615763 

3 

.000739098 

18 

.000705219 

3 

.000674309 

8 

000645995 

6 

000615006 

4 

.000738552 

19  000704722 

4 

.000673854 

9 

000645578 

8 

000614250 

5 

.000738007 

1  420  1.  000704225 

5 

.000673401 

1550 

.000645161 

630 

000613497 

MATHEMATICAL  TABLES. 


No. 
"1632 

Recipro- 
cal* 

No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

No. 

Recipro- 
cal. 

.000612745 

1706 

.000586166 

1780 

.000561798 

1854 

.000539374 

1928 

.000518672 

4 

.00061  1995 

8 

.000585480 

2 

000561167 

6 

.000538793 

1930 

.000518135 

6 

.00061  1247 

1710 

.000584795 

4 

.000560538 

8 

.000538213 

2 

.000517599 

8 

.000610500 

12 

.0005841  12 

6 

.000559910 

1860 

.000537634 

4 

.000517063 

1640 

.000609756 

14 

.000583430 

8 

000559284 

2 

.000537057 

6 

.000516528 

2 

.000609013 

16 

.000582750 

1790 

.000558659 

4 

.000536480 

8 

.0005159% 

4 

.000608272 

18 

000582072 

2 

.000558035 

6 

.000535905 

1940 

.000515464 

6 

.000607533 

1720 

.000581395 

4 

.000557413 

8 

.000535332 

2 

.000514933 

8 

.0006067% 

2 

.000580720 

6 

.000556793 

1870 

.000534759 

4 

.000514403 

1650 

.000506061 

4 

.000580046 

8 

.000556174 

2 

.000534188 

6 

.000513874 

2 

.000605327 

6 

.000579374 

1800 

.000555556 

4 

.000533618 

8 

.000513347 

4 

.000604595 

8 

.000578704 

'2 

000554939 

6 

000533049 

1950 

.000512820 

6 

.000503865 

1730 

.000578035 

4 

.000554324 

8 

.000532481 

2 

.000512295 

8 

.000603136 

2 

.000577367 

6 

.000553710 

1880 

.000531915 

4 

.000511770 

1660 

.000602110 

4 

.000576701 

8 

.000553097 

2 

.000531350 

6 

.000511247 

2 

.000601585 

6 

.000576037 

1810 

.000552486 

4 

.000530785 

8 

.000510725 

4 

.000500962 

8 

.000575374 

12 

.000551876 

6 

.000530222 

1960 

.000510204 

6 

.000600240 

1740 

000574713 

14 

.000551268 

8 

000529661 

2 

.000509684 

8 

.000599520 

2 

.000574053 

16 

.000550661 

1890 

.000529100 

4 

.000509165 

1670 

.000598802 

4 

.000573394 

18 

.000550055 

2 

.000528541 

6 

.000508647 

2 

.000598086 

6 

.000572737 

1820 

.000549451 

4 

.000527983 

8 

.000508130 

4 

.000597371 

8 

.000572082 

2 

.000548848 

6 

.000527426 

197C 

.000507614 

6 

.000596658 

1750 

.000571429 

4 

.000548246 

8 

.000526870 

2 

.000507099 

8 

.000595947 

2 

.000570776 

6 

.000547645 

1900 

.000526316 

4 

.000506585 

1680 

.000595238 

4 

.000570125 

8 

.000547046 

2 

.000525762 

6 

.000506073 

2 

000594530 

6 

000569476 

1830 

000546448 

4 

000525210 

8 

.000505561 

4 

.000593824 

8 

.000568828 

2 

.000545851 

6 

.000524659 

1980 

.000505051 

6 

.000593120 

1760 

.000568182 

4 

.000545256 

8 

.000524109 

2 

.000504541 

8 

.000592417 

2 

.000567537 

6 

.000544662 

1910 

.000523560 

4 

.000504032 

1690 

.000591716 

4 

.000566893 

8 

.000544069 

12 

.000523012 

6 

.000503524 

2 

.000591017 

6 

.000566251 

1840 

000543478 

14 

.000522466 

8 

.000503018 

4 

.000590319 

8 

.00056561  1 

2 

.000542888 

16 

.000521920 

1990 

.000502513 

6 

.000589622 

1770 

.000564972 

4 

.000542299 

18 

.000521376 

2 

.000502008 

8 

.000588928 

2 

.000564334 

6 

.000541711 

1920 

.000520833 

4 

.000501504 

1700 

.000588235 

4 

.000563698 

8 

.000541125 

2 

.000520291 

6 

.000501002 

2 

.000587544 

6 

.000563063 

1850 

.000540540 

4 

.000519750 

8 

.000500501 

4 

.000586854 

8 

.000562430 

2 

.000539957 

6 

.000519211 

2000 

.000500000 

Use  of  reciprocals.  —  Reciprocals  may  be  conveniently  used  to  facili- 
tate computations  in  long  division.  Instead  of  dividing  as  usual,  multiply 
the  dividend  by  the  reciprocal  of  the  divisor.  The  method  is  especially 
useful  when  many  different  dividends  are  required  to  be  divided  by  the 
same  divisor.  In  this  case  find  the  reciprocal  of  the  divisor,  and  make  a 
small  table  of  its  multiples  up  to  9  times,  and  use  this  as  a  multiplication- 
table  instead  of  actually  performing  the  multiplication  in  each  case. 

EXAMPLE.  —  9871  and  several  other  numbers  are  to  be  divided  by  1638. 
The  reciprocal  of  1638  is  .000610500. 
Multiples  of  the 


reciprocal: 


1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 


.0006105 
.0012210 
.0018315 
.0024420 
.0030525 
.0036630 
.0042735 
.0048840 
.0054945 
.0061050 


The  table  of  multiples  is  made  by  continuous  addi- 
tion of  6105.     The  tenth  line  is  written  to  check  the 
accuracy  of  the  addition,  but  it  is  not  afterwards  used. 
Operation. • 

Dividend         9871 

Take  from  table  1 0006105 

7 0.042735 

8 00.48840 

9 005.4945 


Quotient 6.0262455 

Correct  quotient  by  direct  division 6.0262515 

The  result  will  generally  be  correct  to  as  many  figures  as  there  are  signi- 
ficant figures  in  the  reciprocal,  less  one,  and  the  error  of  the  next  figure  will 
in  general  not  exceed  one.  In  the  above  example  the  reciprocal  has  six 
significant  figures,  610500,  and  the  result  is  correct  to  five  places  of  figures. 


SQUARES,   CUBES,   SQUARE  AND   CUBE  ROOTS.       03 


SQUARES,  CUBES,  SQUARE  BOOTS  AND  CUBE  ROOTS  OF 
NUMBERS  FROM  0.1  TO   1600. 


No. 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

No. 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

0  1 

.01 

.001 

.3162 

.4642 

3.1 

9.61 

29.791 

.761 

1.458' 

.15 

.0225 

.0034 

.3873 

.5313 

.2 

10.24 

32.768 

.789 

1.474 

.2 

.04 

.008 

.4472 

.5848 

.3 

10.89 

35.937 

.817 

1.489 

.25 

.0625 

.0156 

.500 

.6300 

.4 

11.56 

39.304 

.844 

1.504 

.3 

.09 

.027 

.5477 

.6694 

.5 

12.25 

42.875 

.871 

1.518 

.35 

.1225 

.0429 

.5916 

.7047 

.6 

12.96 

46.656 

.897 

1.533 

.4 

16 

.064 

.6325 

.7368 

.7 

13.69 

50.653 

.924 

1.547 

.45 

.2025 

.0911 

.6708 

.7663 

.8 

14.44 

54.872 

.949 

1.560 

.5 

.25 

.125 

.7071 

.7937 

.9 

15.21 

59.319 

.975 

1.574 

.55 

.3025 

.1664 

.7416 

.8193 

4. 

16. 

64. 

2. 

1.5874 

.6 

.36 

.  .216 

.7746 

.8434 

.1 

16.81 

68.921 

2.025 

1.601 

.65 

.4225 

.2746 

.8062 

.8662 

.2 

17.64 

74.088 

2.049 

1.613 

.7 

.49 

.343 

.8367 

.8879 

.3 

18.49 

79.507 

2.074 

1.626 

.75 

.5625 

.4219 

.8660 

.9086 

.4 

19.36 

85..184 

2.098 

1.639 

.8 

.64 

.512 

.8944 

.9283 

.5 

20.25 

91.125 

2.121 

1.651 

.85 

.7225 

.6141 

.9219 

.9473 

.6 

21.16 

97.336 

2.145 

1.663 

.9 

.81 

.729 

.9487 

.9655 

.7 

22.09 

103.823 

2.168 

1.675 

.95 

.9025 

.8574 

.9747 

.9830 

.8 

23.04 

110.592 

2.191 

1.687 

1. 

1. 

.9 

24.01 

1  1  7.649 

2.214 

1.698 

1.05 

'.1025 

J58 

!025 

1.016 

5. 

25. 

125. 

2.2361 

1.7100 

j 

.21 

.331 

.049 

1.032 

.1 

26.01 

132.651 

2.258 

1.721 

J5 

.3225 

.521 

.072 

1.048 

.2 

27.04 

140.608 

2.280 

1.732 

.2 

.44 

728 

.095 

1.063 

.3 

28.09 

148.877 

2.302 

1.744 

.25 

•  .5625 

.953 

.118 

1.077 

.4 

29.16 

157.464 

2.324 

1.754 

.3 

.69 

2.197 

.140 

1.091 

.5 

30.25 

166.375 

2.345 

1.765 

.35 

.8225 

2.460 

.162 

1.105 

.6 

31.36 

175.616 

2.366 

1.776 

.4 

.96 

2.744 

.183 

1.119 

.7 

32.49 

185.193 

2.387 

1.786 

.45 

2.1025 

3.049 

.204 

1.132 

.8 

33.64 

195.112 

2.408 

1.797 

.5 

2.25 

3.375 

.2247 

1.1447 

.9 

34.81 

205.379 

2.429 

1.807 

.55 

2.4025 

3.724 

.245 

1.157 

6. 

36. 

216. 

2.4495 

1.8171 

.6 

2.56 

4.096 

.265 

1.170 

.1 

37.21 

226.981 

2.470 

1.827 

.65 

2.7225 

4.492 

.285 

K182 

.2 

38.44 

238.328 

2.490 

1.837 

.7 

2.89 

4.913 

.304 

1.193 

.3 

39.69 

250.047 

2.510 

1.847 

.75 

3.0625 

5.359 

.323 

1.205 

.4 

40.96 

262.144 

2.530 

1.857 

.8 

3.24 

5.832 

.342 

1.216 

.5 

42.25 

274.625 

2.550 

1.866 

1.85 

3.4225 

6.332 

.360 

1.228 

.6 

43.56 

287.496 

2.569 

1.876 

1.9 

3.61 

6.859 

.378 

1.239 

.7 

44.89 

300.763 

2.588 

1.885 

1.95 

3.8025 

7.415 

.396 

1.249 

.8 

46.24 

314.432 

2.608 

1.895 

2. 

4. 

8. 

.4142 

1  .2599 

.9 

47.61 

328.509 

2.627 

1.904 

.1 

4.41 

9.261 

.449 

1.281 

7. 

49. 

343*. 

2.6458 

1.9129 

.2 

4.84 

10.648 

.483 

1.301 

j 

50.41 

357.911 

2.665 

1.922 

.3 

5.29 

12.167 

.517 

1.320 

\2 

51.84 

373.248 

2.683 

1.931 

.4 

5.76 

13.824 

.549 

1.339 

.3 

53.29 

389.017 

2.702 

1.940 

.5 

6.25 

15.625 

.581 

1.357 

.4 

54.76 

405.224 

2.720 

1.949 

.6 

6.76 

17.576 

.612 

1.375 

.5 

56.25 

421.875 

2.739 

1.957 

.7 

7.29 

19.683 

.643 

1.392 

.6 

57.76 

438.976 

2.757 

1.966 

.8 

7.84 

21.952 

.673 

1.409 

.7 

59.29 

456.533 

2.775 

1.975 

.9 

8.41 

24.389 

.703 

1.426 

.8 

60.84 

474.552 

2.793 

1.983 

3. 

• 

9. 

27. 

.7321 

1  .4422 

.9 

62.41 

493.039 

2.811 

1.992 

94 


MATHEMATICAL  TABLES. 


No. 

Square 

Cube. 

Sq. 
Root. 

Cube 
Root. 

No. 

Square 

Cube. 

Sq. 
Root. 

Cube 
Root, 

sT~ 

64. 

512. 

2.8284 

2. 

45 

2025 

91123 

6.7082 

3.5569 

65.61 

531.441 

2.846 

2.008 

46 

2116 

97336 

6.7823 

3.5830 

\2 

67.24 

551.368 

2.864 

2.017 

47 

2209 

103823 

6.8557 

3.6088 

.3 

68.89 

571.787 

2.881 

2.025 

48 

2304 

110592 

6.9282 

3.6342 

.4 

70.56 

592.704 

2.898 

2.033 

49 

2401 

1  1  7649 

7. 

3.6593 

.5 

72.25 

614.125 

2.915 

2.041 

50 

2500 

125000 

7.0711 

3.6840 

.6 

73.96 

636.056 

2.933 

2.049 

51 

2601 

132651 

7.1414 

3.7084 

.7 

75.69 

658.503 

2.950 

2.057 

52 

2704 

140608 

7.2111 

3.7325 

.8 

77.44 

681.472 

2.966 

2.065 

53 

2809 

148877 

7.2801 

3.7563 

.9 

79.21 

704.969 

2.983 

2.072 

54 

2916 

]  57  464 

7.3485 

3.7798 

9. 

81. 

729. 

3. 

2.0801 

55 

3025 

166375 

7.4162 

3.8030 

.1 

82.81 

753.571 

3.017 

2.088 

56 

3136 

175616 

7.4833 

3.8259 

.2 

84.64 

778.688 

3.033 

2.095 

57 

3249 

185193 

7.5498 

3.8485 

.3 

86.49 

804.357 

3.050 

2.103 

58 

3364 

195112 

7.6158 

3.8709 

.4 

88.36 

830.584 

3.066 

2.110 

59 

3481 

205379 

7.6811 

3.8930 

.5 

90.25 

857.375 

3.082 

2.118 

60 

3600 

216000 

7.7460 

3.9149 

.6 

92.16 

884.736 

3.098 

2.125 

61 

3721 

226981 

7.8102 

3.9365 

.7 

94.09 

912.673 

3.114 

2.133 

62 

3844 

238328 

7.8740 

3.9579 

.8 

96.04 

941.192 

3.130 

2.140 

63 

3969 

250047 

7.9373 

3.9791 

.9 

98.01 

970.299 

3.146 

2.147 

64 

4096 

262144 

8. 

4. 

10 

100 

1000 

3.1623 

2.1544 

65 

4225 

274625 

8.0623 

4.0207 

11 

121 

1331 

3.3166 

2.2240 

66 

4356 

287496 

8.1240 

4.0412 

12 

144 

1728 

3.4641 

2.2894 

67 

4489 

300763 

8.1854 

4.0615 

13 

169 

2197 

3.6056 

2.3513 

68 

4624 

314432 

8.2462 

4.0817 

14 

196 

2744 

3.7417 

2.4101 

69 

4761 

328509 

8.3066 

4.1016 

15 

225 

3375 

3.8730 

2.4662 

70 

4900 

343000 

8.3666 

4.1213 

16 

256 

4096 

4. 

2.5198 

71 

5041 

357911 

8.4261 

4.1408 

17 

289 

4913 

4.1231 

2.5713 

72 

5184 

373248 

8.4853 

4.1602 

18 

324 

5832 

4.2426 

2.6207 

73 

5329 

389017 

8.5440 

4.1793 

19 

361 

6859 

4.3589 

2.6684 

74 

5476 

405224 

8.6023 

4.1983 

20 

400 

8000 

4.4721 

2.7144 

75 

5625 

421875 

8.6603 

4.2172 

21 

441 

9261 

4.5826 

2.7589 

76 

5776 

438976 

8.7178 

4.2358 

22 

484 

10648 

4.6904 

2.8020 

77 

5929 

456533 

8.7750 

4.2543 

23 

529 

12167 

4.7958 

2.8439 

78 

6084 

474552 

8.8318 

4.2727 

24 

576 

13824 

4.8990 

2.8845 

79 

6241 

493039 

8.8882 

4.2908 

25 

625 

15625 

5. 

2.9240 

80 

6400 

5.12000 

8.9443 

4.3089 

26 

676 

17576 

5.0990 

2.9625 

81 

6561 

531441 

9. 

4.3267 

27 

729 

19683 

5.1962 

3. 

82 

6724 

551368 

9.0554 

4.3445 

28 

784 

21952 

5.2915 

3.0366 

83 

6889 

571787 

9.1104 

4.3621 

29 

841 

24389 

5.3852 

3.0723 

84 

7056 

592704 

9.1652 

4.3795 

30 

900 

27000 

5.4772 

3.1072 

85 

7225 

614125 

9.2195 

4.3968 

31 

961 

29791 

5.5678 

3.1414 

86 

7396 

636056 

9.2736 

4.4140 

32 

024 

32768 

5.6569 

3.1748 

87 

7569 

658503 

9.3276 

4.4310 

33 

089 

35937 

5.74^6 

3.2075 

88 

7744 

681472 

9.3808 

4.4480 

34 

156 

39304' 

5.8310 

3.2396 

89 

7921 

704969 

9.4340 

4.4647 

35 

225 

42875 

5.9161 

3.2711 

90 

8100 

729000 

9.4868 

4.4814 

36 

296 

46656 

6. 

3.3019 

91 

8281 

753571 

9.5394 

4.4979 

37 

369 

50653 

6.0828 

3.3322 

92 

8464 

778688 

9.5917 

4.5144 

38 

444 

54872 

6.1644 

3.3620 

93 

8649 

804357 

9.6437 

4.5307 

39 

521 

59319 

6.2450 

3.3912 

94 

8836 

830584 

9.6954 

4.5468 

40 

600 

64000 

6.3246 

3.4200 

95 

9025 

857375 

9.7468 

4.5629 

41 

681 

68921 

64031 

3.4482 

96 

9216 

884736 

9.7980 

4.5789 

42 

764 

74088 

6.4807 

3.4760 

97 

9409 

912673 

98489 

4.5947 

43 

849 

79507 

6.5574 

3.5034 

98 

9604 

941  192 

9.8995 

4.6104 

44 

936 

85184 

6.6332 

3.5303 

99 

9801 

970299j 

9.9499 

4.6261 

SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.   95 


No. 

Sq. 

Cube 

Sq. 

Root. 

Cube 
Root. 

No. 

Square. 

Cube.. 

Sq. 
Root. 

Cube 
Root. 

Too" 

10000 

1000000 

10. 

4.6416 

155 

24025 

3723875 

12.4499 

5.3717 

101 

10201 

1030301 

10.0499 

4.6570 

156 

24336 

3796416 

12.4900 

5.3832 

102 

10404 

1061208 

10.0995 

4.6723 

157 

24649 

3869893 

12.5300 

5.3947 

103 

10609 

1092727 

10.1489 

4.6875 

158 

24964 

3944312 

12.5698 

5  4061 

104 

10816 

1124864 

10.1980 

4.7027 

159 

25281 

4019679 

12.6095 

5.4175 

105 

11025 

1157625 

10.2470 

4.7177 

160 

25600 

4096000 

12.6491 

5.4288 

106 

11236 

1191016 

10.2956 

4.7326 

161 

25921 

4173281 

12.6886 

5.4401 

107 

11449 

1225043 

10.3441 

4.7475 

162 

26244 

425  1  528 

12.7279 

5.4514 

10S 

11664 

1259712 

10.3923 

4.7622 

163 

26569 

4330747 

12.7671 

5.^26 

109 

11881 

1295029 

10.4403 

4.7769 

164 

26896 

4410944 

12.8062 

5.W37 

110 

12100 

1331000 

10.4881 

4.7914 

165 

27225 

4492125 

12.8452 

5.4848 

1  1  1 

12321 

1367631 

10.5357 

4.8059 

166 

27556 

4574296 

12.8841 

5.4959 

112 

12544 

1404928 

10.5830 

4.8203 

167 

27889 

4657463 

12.9228 

5.5069 

113 

12769 

1442897 

10.6301 

4.8346 

168 

28224 

4741632 

12.9615 

5.5178 

114 

12996 

1481544 

10.6771 

4.8488 

169 

28561 

4826809 

13.0000 

5.5288 

115 

13225 

1  520875 

10.7238 

4.8629 

170 

28900 

4913000 

13.0384 

5.5397 

116 

13456 

1560896 

10.7703 

4.8770 

171 

29241 

5000211 

13.0767 

5.5505 

117 

13689 

1601613 

10.8167 

4.8910 

172 

29584 

5088448 

13.1149 

5.5613 

118 

13924 

1643032 

10.8628 

4.9049 

173 

29929 

5177717 

13.1529 

5.5721 

119 

14161 

1685159 

10.9087 

4.9187 

174 

30276 

5268024 

13.1909 

5.5828 

120 

14400 

1728000 

10.9545 

4.9324 

175 

30625 

5359375 

13.2288 

5.5934 

121 

14641 

1771561 

1  1  .0000 

4.9461 

176 

30976 

5451776 

13.2665 

5.6041 

122 

14884 

1815848 

1  1  .0454 

4.9597 

177 

31329 

5545233 

13.3041 

5.6147 

123 

15129 

1860867 

1  1  .0905 

4.9732 

178 

31684 

5639752 

13.3417 

5.6252 

124 

15376 

1906624 

11.1355 

4.9866 

179 

32041 

5735339 

13.3791 

5.6357 

125 

15625 

1953125 

11.1803 

5.0000 

180 

32400 

5832000 

13.4164 

5.6462 

126 

15876 

2000376 

11.2250 

5.0133 

181 

32761 

5929741 

13.4536 

5.6567 

127 

16129 

2048383 

1  1  .2694 

5.0265 

182 

33124 

6028568 

13.4907 

5.6671 

12S 

16384 

2097152 

11.313" 

5.0397 

183 

33489 

6128487 

13.5277 

5.6774 

129 

16641 

2146689 

11.3578 

5.0528 

184 

33856 

6229504 

13.5647 

5.6873 

130 

16900 

2197000 

11.4018 

5.0658 

185 

34225 

6331625 

13.6015 

5.6980 

131 

17161 

2248091 

11.4455 

5.0788 

186 

34596 

6434856 

13.6382 

5.7083 

132 

17424 

2299963 

11.4891 

5.0916 

187 

34969 

6539203 

13.6748 

5.7185 

133 

17689 

2352637 

11.5326 

5.1045 

188 

35344 

6644672 

13.7113 

5.7287 

134 

17956 

2406104 

11.5758 

5.1172 

189 

35721 

6751269 

13.7477 

5.7388 

135 

18225 

2460375 

11.6190 

5.1299 

190 

36100 

6859000 

13.7840 

5.7489 

136 

18496 

2515456 

11.6619 

5.1426 

191 

36481 

6967871 

13.8203 

5.7590 

137 

18769 

2571353 

11.7047 

5.1551 

192 

36864 

7077888 

13.8564 

5.7690 

138 

19044 

2628072 

11.7473 

5.1676 

193 

37249 

7189057 

13.8924 

5.7790 

139 

19321 

2685619 

11.7898 

5.1801 

194 

37636 

7301384 

13.9284 

5.7890 

140 

19600 

2744000 

11.8322 

5.1925 

195 

38025 

7414875 

13.9642 

5.7989 

141 

19331 

2803221 

11.8743 

5.2048 

196 

38416 

7529536 

14.0000 

5.8088 

142 

20164 

2863288 

11.9164 

5.2171 

1.97 

38809 

7645373 

14.0357 

5.8186 

143 

20449 

2924207 

1  1  .9583 

'5.2293 

198 

39204 

7762392 

14.0712 

5.8285 

144 

20736 

2985984 

12.0000 

5.2415 

199 

39601 

7880599 

14.1067 

5.8383 

145 

21025 

3048625 

120416 

5.2536 

200 

40000 

8000000 

14.1421 

5.8480 

146 

21316 

3112136 

12.0830 

5.2656 

201 

40401 

8120601 

14.1774 

5.8578 

147 

21609 

3176523 

12.1244 

5.2776 

202 

40804 

8242408 

14.2127 

5.8675 

148 

21904 

3241792 

12  1655 

5.2896 

203 

41209 

8365427 

14.2478 

5.8771 

149 

22201 

3307949 

12.2066 

5.3015 

204 

41616 

8489664 

14.2829 

5.8868 

150 

22500 

3375000 

12.2474 

5.3133 

205 

42025 

8615125 

14.3178 

5.8964 

151 

22801 

344295  1 

12.2882 

5.3251 

206 

42436 

8741816 

14.3527 

5.9059 

152 

23104 

3511808 

12.3288 

5.3368 

207 

42849 

8869743 

14.3875 

5.9155 

153 

23409 

3581577 

12.3693 

5.3485 

208 

43264 

8998912 

14.4222 

5.9250 

154  23716 

3652264 

12.4097 

5.3601 

209 

43681 

9129329 

14.4568 

5.9345 

96 


MATHEMATICAL  TABLES. 


No. 

Sq. 

Cube. 

Root. 

Cube 
Root. 

No. 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

2H)" 

44100 

9261000 

14.4914 

5.9439 

265 

70225 

18609625 

16.2788 

6.4232 

211 

44521 

939393  1 

14.5258 

5.9533 

266 

70756 

18821096 

16.3095 

6.4312 

212 

44944 

9528128 

14.5602 

5.9627 

267 

71289 

19034163 

16.3401 

6.4393 

213 

45369 

9663597 

14.5945 

5.9721 

268 

71824 

19248832 

16.3707 

6.4473 

214 

45796 

9800344 

14.6287 

5.9814 

269 

72361 

19465109 

16.4012 

6.4553 

215 

46225 

9938375 

14.6629 

5.9907 

270 

72900 

19683000 

16.4317 

6.4633 

216 

46656 

10077696 

14.6969 

6.0000 

271 

73441 

19902511 

16.4621 

6.4713_ 

217 

47089 

1  02  1  83  1  3 

14.7309 

6.0092 

272 

73984 

20123648 

16.4924 

6.4792 

21S 

47524 

10360232 

14.7648 

6.0185 

273 

74529 

20346417 

16.5227 

6.4872 

1 

47961 

10503459 

14.7986 

6.0277 

274 

75076 

20570824 

16.5529 

6.4951 

220 

48400 

10648000 

14.8324 

6.0368 

275 

75625 

20796875 

16.5831 

6.5030 

221 

48841 

10793861 

14.8661 

6.0459 

276 

76176 

21024576 

16.6132 

6.5108 

222 

49284 

1  094  1  048 

14.8997 

6.0550 

277 

76729 

21253933 

16.6433 

6.5187 

223 

49729 

11089567 

14.9332 

6.0641 

278 

77284 

21484952 

16.6733 

6.5265 

224 

50176 

11239424 

14.9666 

6.0732 

279 

77841 

21717639 

16.7033 

6.5343 

225 

50625 

11390625 

15.0000 

6.0822 

280 

78400 

21952000 

16.7332 

6.5421 

226 

51076 

11543176 

15.0333 

6.0912 

281 

78961 

22188041 

16.7631 

6.5499 

227 

51529 

11697083 

15.0665 

6.1002 

282 

79524 

22425768 

16.7929 

6.5577 

228 

51984 

11852352 

15.0997 

6.1091 

283 

80089 

22665187 

16.8226 

6.5654 

229 

52441 

12008989 

15.1327 

6.1180 

284 

80656 

22906304 

16.8523 

6.5731 

230 

52900 

12167000 

15.1658 

6.1269 

285 

81225 

23149125 

16.8819 

6.5808 

231 

53361 

12326391 

15.1987 

6.1358 

286 

81796 

23393656 

16.9115 

6.5885 

232 

53824 

12487168 

15.2315 

6.  1  446 

287 

82369 

23639903 

16.9411 

6.5962 

233 

54289 

12649337 

15.2643 

6.1534 

288 

82944 

23887872 

16.9706 

6.6039 

234 

54756 

12812904 

15.2971 

6.1622 

289 

83521 

24137569 

17.0000 

6.6115 

235 

55225 

12977875 

15.3297 

6.1710 

290 

84100 

24389000 

17.0294 

6.6191 

236 

55696 

13144256 

15.3623 

6.1797 

291 

84681 

24642171 

17.0587 

6.6267 

237 

56169 

13312053 

153948 

6.1885 

292 

85264 

24897088 

17.0880 

6.6343 

238 

56644 

13481272 

15.4272 

6.1972 

293 

85849 

25153757 

17.1172 

6.6419 

239 

57121 

13651919 

15.4596 

6.2058 

294 

86436 

25412184 

17.1464 

6.6494 

240 

57600 

13824000 

15.4919 

6.2145 

295 

87025 

25672375 

17.1756 

6.6569 

241 

58081 

13997521 

15.5242 

6.223  1 

296 

87616 

25934336 

17.2047 

6.6644 

242 

58564 

14172488 

15.5563 

6.2317 

297 

88209 

26198073 

17.2337 

6.6719 

243 

59049 

14348907 

15.5885 

6.2403 

298 

88804 

26463592 

17.2627 

6.6794 

244 

59536 

14526784 

15.6205 

6.2488 

299 

89401 

26730899 

17.2916 

6.6869 

245 

60025 

14706125 

15.6525 

6.2573 

300 

90000 

27000000 

17.3205 

6.6943 

246 

60516 

14886936 

15.6844 

6.2658 

301 

90601 

27270901 

17.3494 

6.7018 

247 

61009 

1  5069223 

15.7162 

6.2743 

302 

91204 

27543608 

17.3781 

6.7092 

248 

61504 

15252992 

15.7480 

6.2828 

303 

91809 

27818127 

17.4069 

6.7166 

249 

62001 

15438249 

15.7797 

6.2912 

304 

92416 

28094464 

17.4356 

6.7240 

250 

62500 

1  5625000 

15.8114 

6.2996 

305 

93025 

28372625 

1  7.4642 

6.7313 

251 

63001 

15813251 

15.8430 

6.3080 

306 

93636 

28652616 

17.4929 

6.7387 

252 

63504 

16003008 

15.8745 

6.3164 

307 

94249 

28934443 

17.5214 

6.7460 

253 

64009 

16194277 

15.9060 

6.3247 

308 

•  94864 

29218112 

17.5499 

6.7533 

254 

64516 

16387064 

15.9374 

6.3330 

309 

95481 

29503629 

17.5784 

6.7606 

255 

65025 

16581375 

15.9687 

6.3413 

310 

96100 

29791000 

1  7.6068 

6.7679 

256 

65536 

16777216 

16.0000 

6.3496 

311 

96721 

3008023  1 

17.6352 

6.7752 

257 

66049 

16974593 

16.0312 

6,3579 

312 

97344 

30371328 

17.6635 

6.7824 

258 

66564 

17173512 

16.0624 

6.3661 

313 

97969 

30664297 

17.6918 

6.7897 

259 

67081 

17373979 

16.0935 

6.3743 

314 

98596 

30959144 

17.7200 

6.7969 

260 

67600 

17576000 

16.1245 

6.3825 

315 

99225 

31255875 

17.7482 

6.8041 

261 

68121 

17779581 

16.1555 

63907 

316 

99856 

31554496 

17.7764 

6.8113 

262 

68644 

1  7984728 

16.1864 

6.3988 

317 

100489 

31855013 

17.8045 

6.8185 

263 

69169 

18191447 

16.2173 

6.4070 

318 

101124 

32157432 

17.8326 

6.8256 

264 

69696 

18399744  16.2481 

6.4151 

319 

101761 

32461759 

17.8606 

6.8328 

SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.   97 


No. 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

No. 

Square 

Cube. 

Sq. 
Root. 

Cube 
Root. 

320 

102400 

32768000 

17.8885 

6.8399 

375 

1  40625 

52734375 

19.3649 

7.2112 

321 

103041 

33076161 

17.9165 

6.8470 

376 

141376 

53157376 

19.3907 

7.2177 

322 

1  03684 

33386248 

17.9444 

6.8541 

377 

142129 

53582633 

19.4165 

7.2240 

323 

104329 

33698267 

17.9722 

6.8612 

378 

1  42884 

54010152 

1  9.4422 

7.2304 

324 

104976 

34012224 

18.0000 

6.8683 

379 

143641 

54439939 

19.4679 

7.2368 

325 

105625 

34328125 

18.0278 

6.8753 

380 

1  44400 

54872000 

1  9.4936 

7.2432 

326 

106276 

34645976 

18.0555 

6.8824 

381 

145161 

55306341 

19.5192 

72495 

327 

106929 

34965783 

18.0831 

6.8894 

382 

145924 

55742968 

19.5448 

7.2558 

328 

107584 

35287552 

18.1108 

6.8964 

383 

1  46689 

56181887 

19.5704 

7.2622 

329 

108241 

35611289 

18.1384 

6.9034 

384 

147456 

56623104 

19.5959 

7.2685 

330 

108900 

35937000 

18.1659 

6.9104 

385 

148225 

57066625 

19.6214 

7.2748 

331 

109561 

36264691 

18.1934 

6.9174 

386 

1  48996 

57512456 

19.6469 

7.2811 

332 

110224 

36594368 

18.2209 

6.9244 

387 

149769 

57960603 

19.6723 

7.2874 

333 

110889 

36926037 

18.2483 

6.9313 

388 

150544 

58411072 

19.6977 

7.2936 

334 

111556 

37259704 

18.2757 

6.9382 

389 

151321 

58863869 

19.7231 

7.2999 

335 

112225 

37595375 

18.3030 

6.9451 

390 

152100 

59319000 

19.7484 

7.306! 

336 

1  1  2896 

37933056 

18.3303 

6.9521 

391 

152881 

59776471 

19.7737 

7.3124 

337 

113569 

38272753 

183576 

6.9589 

392 

153664 

60236288 

19.7990 

7.3186 

338 

114244 

38614472 

18.3848 

6.9658 

393 

1  54449 

60698457 

1  9.8242 

7.3248 

339 

114921 

38958219 

18.4120 

6.9727 

394 

155236 

61162984 

1  9.8494 

7.3310 

340 

1  1  5600 

39304000 

18.4391 

6.9795 

395 

156025 

61629875 

19.8746 

7.3372 

341 

116281 

39651821 

18.4662 

6.9864 

396 

156816 

62099136 

1  9.8997 

7.3434 

342 

116964 

40001688 

18.4932 

6.9932 

397 

157609 

62570773 

19.9249 

7.3496 

343 

1  1  7649 

40353607 

18.521)3 

7.0000 

398 

1  58404 

63044792 

19.9499 

7.35*8 

344 

118336 

40707584 

18.5472 

7.0068 

399 

159201 

63521199 

19.9750 

7.361$ 

345 

119025 

41063625 

18.5742 

7.0136 

400 

1  60000 

64000000 

20.0000 

7.3681 

346 

119716 

41421736 

18.6011  7.0203 

401 

160801 

64481201 

20.0250 

7.3742 

347 

120409 

41781923 

18.62797.0271 

402 

161604 

64964808 

20.0499 

7.3803 

348 

121104 

42144192 

18.6548 

7.0338 

403 

162409 

65450827 

20.0749 

7.3864 

349 

121801 

42508549 

18.6815 

7.0406 

404 

163216 

65939264 

20.0998 

7.3925 

350 

122500 

42875000 

18.7083 

7.0473 

405 

164025 

66430125 

20.1246 

7.3986 

351 

123201 

43243551 

18.7350 

7.0540 

406 

164836 

66923416 

20.1494 

7.4047 

352 

123904 

43614208 

18.7617 

7.0607 

407 

165649 

67419143 

20.1742 

7.4108 

353 

124609 

43986977 

18.7883 

7.0674 

408 

166464 

67917312 

20.1990 

7.4169 

354 

125316 

44361864 

18.8149 

7.0740 

409 

167281 

68417929 

20.2237 

7.4229 

355 

126025 

44738875 

18.8414 

7.0807 

410 

168100 

68921  COO 

202485 

7.4290 

356 

126736 

45118016 

18.8680 

7.0873 

411 

168921 

69426531 

20.2731 

7.4350 

357 

127449 

45499293 

18.8944 

7.0940 

412 

1  697  44 

69934528 

20.2978 

7.4410 

358 

128164 

45882712 

18.9209 

7.1006 

413 

170569 

70444997 

20.3224 

7.4470 

359 

128881 

46268279 

18.9473 

7.1072 

414 

171396 

70957944 

20.3470 

7.4530 

360 

129600 

46656000 

189737 

7.1138 

415 

172225 

71473375 

20.3715 

7.4590 

361 

130321 

47045881 

19.0000 

7.1204 

416 

173056 

71991296 

20.3961 

7.4650 

362 

131044 

47437928 

19.0263 

7.1269 

417 

1  73889 

72511713 

20.4206 

7.4710 

363 

131769 

47832147 

19.0526 

7.1335 

418 

1  74724 

73034632 

20.4450 

7.4770 

364 

132496 

48228544 

19.0788 

7.1400 

419 

175561 

73560059 

20.4695 

7.4829 

365 

133225 

48627125 

19.1050 

7.1466 

420 

1  76400 

74088000 

20.4939 

74889 

366 

133956 

49027896 

19.1311 

7.1531 

421 

177241 

74618461 

20.5183 

7.4948 

367 

134689 

49430863 

19.1572 

7.1596 

422 

1  78084 

75151448 

20.5426 

7.5007 

368 

135424 

49836032 

19.1833 

7.1661 

423 

1  78929 

75686967 

20.5670 

7.5067 

369 

136161 

50243409 

19.2094 

7.1726 

424 

179776 

76225024 

20.5913 

7.5126 

370 

1  36900 

50653000 

19.2354 

7.1791 

425 

180625 

76765625 

20.6155 

7.5185 

371 

137641 

51064811 

19.2614 

7.1855 

426 

18M76 

77308776 

20  6398 

7  5244 

372 

138384 

51478848 

19.2873 

7.1920 

427 

182329 

77854483 

20.6640 

7  5302 

373 

374 

139129 
139876 

51895117  19313217  1984 
52313624  19.3391  '7.2048 

428 
429 

183184 
184041 

78402752 
78953589 

20.6882 
20.7123 

7.5361 
7.5420 

'98 


MATHEMATICAL  TABLES. 


No. 

Square 

Cube. 

Sq. 
Root. 

Cube 
Root. 

No. 

Square 

Cube. 

Sq. 
Root. 

Cube 
Root. 

430 
431 
432 
433 
434 

184900 
185761 
186624 
187489 
188356 

79507000 
80062991 
80621568 
81182737 
81746504 

20.7364 
20.7605 
20.7846 
20.8087 
20.8327 

7.5478 
7.5537 
7.5595 
7.5654 
7.5712 

485 
486 
487 
488 
489 

235225 
236196 
237169 
238144 
239121 

114084125 
114791256 
115501303 
116214272 
116930169 

22.0227 
22.0454 
22.0681 
22.0907 
22.1133 

7.8568 
7.8622 
7.8676 
7.8730 
7.8784 

435 
436 
437 
438 
439 

189225 
190096 
1  90969 
191844 
192721 

82312875 
82881856 
83453453 
84027672 
846045  1  9 

20.8567 
20.8806 
20.9045 
20.9284 
20.9523 

7.5770 

7.5828 
7.5886 
7.5944 
76001 

490 
491 
492 
493 
494 

240100 
241081 
242064 
243049 
244036 

1  1  7649000 
118370771 
119095488 
119823157 
120553784 

22.1359 
22.1585 
22.1811 
22.2036 
22.2261 

7.8837 
7.8891 
7.8944 
7.8998 
7.9051 

440 
441 
442 
443 
444 

193600 
194481 
195364 
196249 
197136 

85184000 
85766121 
86350888 
86938307 
87528384 

20.9762 
21.0000 
21.0238 
21.0476 
21.0713 

7.6059 
7.6117 
7.6174 
7.6232 
7.6289 

495 
496 
497 
498 
499 

245025 
246016 
247009 
248004 
249001 

121287375 
122023936 
122763473 
123505992 
124251499 

22.2486 
22.2711 
22.2935 
22.3159 
22.3383 

7.9105 
7.9158 
7.9211 
7.9264 
7.9317 

445 
446 
447 
448 
449 

198025 
198916 
199809 
200704 
201601 

88121125 
88716536 
893  1  4623 
89915392 
90518849 

21  0950 
21.1187 
21.1424 
21.1660 
21.1896 

7.6346 
7.6403 
7.6460 
7.6517 
7.6574 

500 
501 
502 
503 
504 

250000 
251001 
252004 
253009 
254016 

125000000 
125751501 
1  26506008 
127263527 
128024064 

22.3607 
22.3830 
22.4054 
22.4277 
22.4499 

7.9370 
7.9423 
7.9476 
7.9528 
7.9581 

450 
451 
452 
453 
454 

202500 
203401 
204304 
205209 
206116 

91125000 
91733851 
92345408 
92959677 
93576664 

21.2132 
21.2368 
21.2603 
21.2838 
21.3073 

7.6631 
7.6688 
7.6744 
7.6800 
7.6857 

505 
506 
507 
508 
509 

255025 
256036 
257049 
258064 
259081 

128787625 
129554216 
130323843 
131096512 
131872229 

22.4722 
22.4944 
22.5167 
22.5389 
22.5610 

7.9634 
7.9686 
79739 
7.9791 
7.9843 

455 
456 
457 
458 
459 

207025 
207936 
208849 
209764 
210681 

94196375 
94818816 
95443993 
96071912 
96702579 

21.3307 
21.3542 
21.3776 
21.4009 
21.4243 

7.6914 
7.6970 
7.7026 
7.7082 
7.7138 

510 
511 
512 
513 
514 

260100 
261121 
262144 
263169 
264196 

132651000 
133432831 
134217728 
135005697 
135796744 

22.5832 
22.6053 
22.6274 
22.6495 
22.6716 

7.9896 
7.9948 
8.0000 
8.0052 
8.0104 

460 
461 
462 
463 
464 

211600 
212521 
213444 
214369 
215296 

97336000 

97972181 
98611128 
99252847 
99897344 

21.4476 
21.4709 
21.4942 
21.5174 
21.5407 

7.7194 
7.7250 
7.7306 
7.7362 
7.7418 

515 
516 
517 
518 
519 

265225 
266256 
267289 
268324 
269361 

136590875 
137388096 
138188413 
138991832 
139798359 

22  6936 
22.7156 
22.7376 
22.7596 
22.7816 

8.0156 
8.0208 
8.0260 
8.0311 
8.0363 

465 
466 
467 
468 
469 

216225 
217156 
218089 
219024 
219961 

100544625 
101194696 
101847563 
102503232 
103161709 

21.5639 
21.5870 
21.6102 
21.6333 
21.6564 

7.7473 
7.7529 
7.7584 
7.7639 
7.7695 

520 
521 
522 
523 
524 

270400 
271441 
272484 
273529 
274576 

140608000 
141420761 
142236648 
143055667 
143877824 

22.8035 
22.8254 
22.8473 
22.8692 
22.8910 

8.0415 
8.0466 
8.0517 
8.0569 
8.0620 

470 
471 
472 
473 
474 

220900 
221841 
222784 
223729 
224676 

103823000 
104487111 
105154048 
105823817 
106496424 

21.6795 
21.7025 
21.7256 
21.7486 
21.7715 

7.7750 
7.7805 
7.7860 
7.7915 
7.7970 

525 
526 
527 

528 
529 

275625 
276676 
277729 

278784 
279841 

144703125 
145531576 
146363183 
147197952 
148035889 

22.9129 
22.9347 
22.9565 
22.9783 
23.0000 

8.0671 
8.0723 
8.0774 
8.0825 
8.0876 

475 
476 
477 
478 
479 

225625 
226576 
227529 
228484 
22944  1 

107171875 
107850176 
108531333 
109215352 
109902239 

21.7945 
21.8174 
21  8403 
21.8632 
21.8861 

7.8025 
7.8079 
7.8134 
7.8188 
7.8243 

530 
531 
532 
533 
534 

280900 
281961 
283024 
284089 
285156 

148877000 
149721291 
1  50568768 
151419437 
152273304 

23.0217 
23.0434 
23.0651 
23  0868 
23.1084 

80927 
8.0978 
8.1028 
8.1079 
8.1130 

480 
481 
482 
483 
484 

230400 
231361 
232324 
233289 
234256 

110592000 
111284641 
111980168 
112678587 
1  13379904 

21.9089 
21.9317 
21.9545 
21.9773 
22.0000 

7.8297 
7.8352 
7.8406 
7.8460 
7.8514 

535 
536 
537 
538 
539 

286225 
287296 
288369 
289444 
290521 

153130375 
1  53990656 
154854153 
155720S72 
156590819 

23.1301 
23.1517 
23.1733 
23.1948 
23.2164 

8.1180 
8.123! 
8.128! 
8.1332 
8.1332 

SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.   99 


No. 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

No. 

Square 

Cube. 

Sq. 
Root. 

Cube 
Root. 

540 
541 
542 
543 
544 

291600 
292681 
293764 
294849 
295936 

157464000 
158340421 
159220088 
160103007 
160989184 

23.2379 
23.2594 
23.2809 
23.3024 
23.3238 

8.1433 
8.1483 
8.1533 
8.1583 
8.1633 

595 

596 
597 
598 
599 

354025 
355216 
356409 
357604 
358801 

210644875 
211708736 
212776173 
213847192 
214921799 

24.3926 
24.4131 
24.4336 
24.4540 
24.4745 

8.4108 
8.4155 
8.4202 
8.4249 
8.4296 

545 

546 
547 
548 
549 

297025 
298116 
299209 
300304 
301401 

161878625 
162771336 
163667323 
164566592 
165469149 

23.3452 
23.3666 
23.3880 
23.4094 
23.4307 

8.1683 
8.1733 
8.1783 
8.1833 
8.1882 

600 
601 
602 
603 
604 

360000 
361201 
362404 
363609 
364816 

216000000 
217081801 
218167208 
219256227 
220348864 

24.4949 
24.5153 
24.5357 
24.5561 
24.5764 

8.4343 
8.4390 
8.4437 
8.4484 
8.4530 

550 
551 
552 
553 
554 

302500 
303601 
304704 
305809 
306916 

166375000 
167284151 
168196608 
169112377 
170031464 

23.4521 
23.4734 
23.4947 
23.5160 
23.5372 

8.1932 
8.1982 
8.2031 
8.2081 
8.2130 

605 
606 
607 
608 
609 

366025 
367236 
36S449 
369664 
370881 

221445125 
222545016 
223648543 
224755712 
225866529 

24.5967 
24.6171 
24.6374 
24.6577 
24.6779 

8.4577 
8.4623 
8.4670 
8.4716 
8.4763 

555 

556 
557 

558 
559 

308025 
309136 
310249 
311364 
312481 

170953875 
171879616 
1  72808693 
173741112 
174676879 

23.5584 
23.5797 
23.6008 
23.6220 
23.6432 

8.2180 
8.2229 
8.2278 
8.2327 
8.2377 

610 
611 
612 
613 
614 

372100 
373321 
374544 
375769 
376996 

226981000 
228099131 
229220928 
230346397 
231475544 

24.6982 
24.7184 
24.7386 
24.7588 
24.7790 

8.4809 
8.4856 
8.4902 
8.4948 
8.4994 

560 
561 
562 
563 
564 

313600 
314721 
315844 
316969 
318096 

175616000 
176558481 
177504328 
178453547 
179406144 

23.6643 
23.6854 
23.7065 
23.7276 
23.7487 

8.2426 
8.2475 
8.2524 
8.2573 
8.2621 

615 
616 
617 
618 
619 

378225 
379456 
380689 
381924 
383161 

232608375 
233744896 
234885113 
236029032 
237176659 

24.7992 
24.8193 
24.8395 
24.8596 
24.8797 

8.5040 
8.5C86 
8.5132 
8.5178 
8.5224 

565 
566 
567 
568 
569 

319225 
320356 
321489 
322624 
323761 

180362125 
181321496 
182284263 
183250432 
184220009 

23.7697 
23.7908 
23.8118 
23.8328 
23.8537 

8.2670 
8.2719 
8.2768 
8.2816 
8.2865 

620 
621 
622 
623 
624 

384400 
385641 
386884 
388129 
389376 

238328000 
239483061 
240641848 
241804367 
242970624 

24.8998 
24.9199 
24.9399 
24.9600 
24.9800 

8.5270 
8.5316 
8.5362 
8.5408 
8.5453 

570 
571 
572 
573 
574 

324900 
326041 
327184 
328329 
329476 

185193000 
186169411 
187149248 
188132517 
189119224 

23.8747 
23.8956 
23.9165 
23.9374 
23.9583 

8.2913 
8.2962 
8.3010 
8.3059 
8.3107 

625 
626 
627 
628 
629 

390625 
391876 
393129 
394384 
395641 

244140625 
245314376 
246491883 
247673152 
248858189 

25.0000 
25.0200 
25.0400 
25.0599 
25.0799 

8.5499 
8.5544 
85590 
8.5635 
8.5681 

575 
576 
577 
578 
579 

330625 
331776 
332929 
334084 
335241 

190109375 
191102976 
192100033 
193100552 
194104539 

23.9792 
24.0000 
24.0208 
24.0416 
24.0624 

8.3155 
8.3203 
8.3251 
8.3300 
8.3348 

630 
631 
632 
633 
634 

396900 
398161 
399424 
400689 
401956 

250047000 
251239591 
252435968 
253636137 
254840104 

25.0998 
25.1197 
25.1396 
25.1595 
25.1794 

8.5726 
8.5772 
8.5817 
8.5862 
8.5907 

580 
581 
582 
583 
584 

336400 
337561 
338724 
339889 
341056 

195112000 
196122941 
197137368 
198155287 
199176704 

24.0832 
24.1039 
24.1247 
24.1454 
24.1661 

8.3396 
8.3443 
8.3491 
8.3539 
8.3587 

635 
636 
637 
638 
639 

403225 
404496 
405769 
407044 
408321 

256047875 
257259456 
258474853 
259694072 
260917119 

25.1992 
25.2190 
25.2389 
25.2587 
25.2784 

8.5952 
8.5997 
8.6043 
8.6088 
8.6132 

585 
586 
587 
588 
589 

342225 
343396 
344569 
345744 
34692  1 

200201625 
201230056 
202262003 
203297472 
204336469 

24.1868 
24.2074 
24.2281 
24.2487 
24.2693 

8.3634 
8.3682 
8.3730 
8.3777 
8.3825 

640 
641 
642 
643 
644 

409600 
410881 
412164 
413449 
414736 

262144000 
263374721 
264609288 
265847707 
267089984 

25.2982 
25.3180 
25.3377 
25.3574 
25.3772 

8.6177 
8.6222 
8.6267 
8.6312 
8.6357 

590 
591 
592 
593 
594 

348100 
349281 
350464 
351649 
352836 

205379000 
206425071 
207474688 
208527857 
209584584 

24.2899 
24.3105 
24.3311 
24.3516 
24.3721 

8.3872 
8.3919 
8.3967 
8.4014 
8.4061 

645 
646 
647 
648 
649 

416025 
417316 
4  1  8609 
419904 
421201 

268336125 
269586136 
27084002.3 
272097792 
273359449 

25.3969 
25.416 
25.436 
25.4558 
25.475 

8.6401 
8.6446 
8.6490 
8.6535 
8.6579 

100 


MATHEMATICAL  TABLES. 


No. 

650 
651 
652 
653 
654 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

No. 

705 
706 
707 
708 
709 

Square 

Cube* 

Sq. 
Root. 

Cube 
Root. 

422500 
42380! 
425104 
426409 
427716 

274625000 
275894451 
277167808 
278445077 
279726264 

25.4951 
25.5147 
25.5343 
25.5539 
25.5734 

6.6624 
8.6668 
8.6713 
8.6757 
8.6801 

497625 
498436 
499849 
501264 
502681 

350402625 
351895816 
353393243 
354894912 
356400829 

26.5518 
26.5707 
26.5895 
26.6083 
26.6271 

8.9001 
8.9043 
8.9085 
8.9127 
8.9169 

655 
656 
657 
658 
659 

429025 
430336 
431649 
432964 
434281 

281011375 
282300416 
283593393 
284890312 
286191179 

25.5930 
25.6125 
25.6320 
25.6515 
25.6710 

8.6845 
8.6890 
8.6934 
8.6978 
8.7022 

710 
711 
712 
713 
714 

504100 
505521 
506944 
508369 
509796 

35791  100C 
359425431 
360944128 
362467097 
363994344 

26.6458 
26.6646 
26.6833 
26.7021 
26.7208 

8.9211 
8.9253 
8.9295 
8.9337 
8.9378 

660 
661 
662 
663 
664 

435600 
43692  1 
438244 
439569 
440896 

287496000 
288804781 
290117528 
291434247 
292754944 

25.6905 
25.7099 
25.7294 
25.7488 
25.7682 

8.7066 
8.7110 
8.7154 
8.7198 
8.7241 

715 
716 
717 
718 
719 

511225 
512656 
5  1  4089 
515524 
516961 

365525875 
367061696 
368601813 
370146232 
371694959 

26.7395 
26.7582 
26.7769 
26.7955 
26.8142 

8.9420 
8.9462 
8.9503 
8.9545 
8.9587 

665 
566 
667 
663 
669 

442225 
443556 
444889 
446224 
447561 

294079625 
295408296 
296740963 
298077632 
299418309 

25.7876 
25.8070 
25.8263 
25.8457 
25.8650 

8.7285 
87329 

8.7373 
8.7<16 
8.7460 

720 

721 
722 
723 
724 

518400 
519841 
521284 
522729 
524176 

373248000 
374805361 
376367048 
377933067 
379503424 

26.8328 
26.8514 
26.8701 
26.8887 
26.9072 

8.9628 
8.9670 
8.9711 
8.9752 
8.9794 

670 
671 
672 
673 
674 

448900 
450241 
451584 
452929 
454276 

300763000 
302111711 
303464448 
304821217 
306182024 

25.8844 
25.9037 
25.9230 
25.9422 
25.9615 

8.7503 
8.7547 
8.7590 
8.7634 
8.7677 

725 
726 
727 
728 
729 

525625 
527076 
528529 
529984 
531441 

381078125 
382657176 
384240583 
385828352 
387420489 

26.9258 
26.9444 
26.9629 
26.9815 
27.0000 

8.9835 
8.9876 
8.9918 
8.9959 
9.0000 

675 
676 
677 
673 
679 

455625 
456976 
458329 
459684 
461041 

307546875 
308915776 
310288733 
311665752 
313046839 

25.9808 
26.0000 
26.0192 
26.0384 
26.0576 

8.7721 
8.7764 
8.7807 
8.7850 
8.7893 

730 
731 

732 
733 
734 

532900 
534361 
535824 
537289 
538756 

389017000 
390617891 
392223168 
393832837 
395446904 

27.0185 
27.0370 
27.0555 
27.0740 
27.0924 

9.0041 
9.0082 
9.0123 
9.0164 
9.0205 

680 
681 
682 
683 
684 

462400 
463761 
465124 
466489 
467856 

314432000 
315821241 
317214568 
318611987 
320013504 

26.0768 
26.0960 
26.1151 
26.1343 
26.1534 

8.7937 
8.7980 
8.8023 
8.8066 
8.8109 

735 

736 
737 
738 
739 

540225 
541696 
543169 
5  4464  4 
546121 

397065375 
398688256 
400315553 
401947272 
403583419 

27.1109 
27.1293 
27.1477 
27.1662 
27.1846 

9.0246 
9.0287 
9.0328 
9.0369 
9.0410 

685 
686 
687 
688 
689 

469225 
470596 
471969 
473344 
474721 

321419125 
322828856 
324242703 
325660672 
327082769 

26.1725 
26.1916 
26.2107 
26.2298 
26.2488 

8.8152 
8.8194 
8.8237 
8.8280 
8.8323 

740 
741 
742 
743 
744 

54760C 
54908  1 
550564 
552049 
553536 

405224000 
406869021 
408518488 
410172407 

411830784 

27.2029 
27.2213 
27.2397 
27.2580 
27.2764 

9.0450 
90491 
9.0532 
9.0572 
9.0613 

690 
691 
692 
693 
694 

476100 
477481 
478864 
480249 
481636 

328509000 
329939371 
331373888 
332812557 
334255384 

26.2679 
26.2869 
26.3059 
26.3249 
26.3439 

8.8366 
8.8408 
8.8451 
8.8493 
8.8536 

745 

746 
747 
748 
749 

555025 
556516 
558009 
559504 
561001 

413493625 
415160936 
416832723 
418508992 
420189749 

27.2947 
273130 
27.3313 
27.3496 
27.3679 

9.0654 
9.0694 
9.0735 
9.0775 
9.0816 

695 
696 
697 
698 
699 

483025 
484416 
485809 
487204 
488601 

335702375 
337153536 
338608873 
340068392 
341532099 

26.3629 
26.3818 
26.4008 
26.4197 
26.4386 

8.8578 
8.8621 
8.8663 
8.8706 
8.8748 

750 
751 
752 
753 

754 

562500 
564001 
565504 
567009 
568516 

421875000 
423564751 
425259008 
426957777 
428661061 

27.3861 
27.4044 
27.4226 
27.4408 
27.4591 

9.0856 
90896 
9.0937 
9.0977 
9.1017 

700 
701 
702 
703 
704 

490000 
491401 
492804 
494209 
495616 

343000000 
344472101 
345948408 
347428927 
348913664 

26.4575 
26.4764 
26.4953 
26.5141 
26.5330 

88790 
8.8833 
8.8875 
8.8917 
8.8959 

755 
756 
757 
758 
759 

570025 
571536 
573049 
574564 
576081 

430368875 
432081216 
433798093 
435519512 
437245479 

27.4773 
27.4955 
27.5136 
27.5318 
27.5500 

9  1057 
9.1098 
9.1138 
9  1178 
9.1219 

SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.       101 


No 

Square 

Cube. 

Sq. 
Root. 

Cube 
Root. 

No 

Square 

Cube. 

Sq. 
Root. 

Cube 
Root. 

760 

577600 

438976000 

27.5681 

9.1258 

~8l5 

664225 

541343375 

28.5482 

9.3408 

76 

57912 

440711081 

27.5862 

9.1298 

816 

665856 

543338496 

28.5657 

9.3447 

762 

580644 

442450728 

27.6043 

9.1338 

81 

667489 

545338513 

28.5832 

9.3485 

763 

532  1  69 

444194947 

27.6225 

9.1378 

818 

66912 

547343432 

28.6007 

9.3523 

764 

583696 

445943744 

27.6405 

9.1418 

819 

67076 

549353259 

28.6182 

9.3561 

765 

585225 

447697125 

27.6586 

9.1458 

820 

672400 

55136800C 

28.6356 

9.3599 

766 

586756 

449455096 

27.6767 

9.  1  498 

82 

67404 

55338766 

28.6531 

9.3637 

767 

588289 

451217663 

27.6948 

9.1537 

822 

67568 

55541224S 

28.6705 

9.3675 

768 

589824 

452984832 

27.7128 

9.1577 

823 

67732 

55744176 

28.6880 

9.3713 

769 

591361 

454756609 

27.7308 

9.1617 

824 

678976 

559476224 

28.7054 

9.3751 

770 

592900 

456533000 

27.7489 

9.1657 

825 

68062 

561515625 

28.7228 

9.3789 

771 

594441 

458314011 

27.7669 

9.1696 

826 

682276 

563559976 

28.7402 

9.3827 

772 

595984 

460099648 

27.7849 

9.1736 

827 

683929 

565609283 

28.7576 

9.3865 

773 

597529 

461889917 

27.8029 

9.1775 

828 

685584 

567663552 

28.7750 

9.3902 

774 

599076 

463684824 

27.8209 

9.1815 

829 

68724 

569722789 

28.7924 

9.3940 

775 

600625 

465434375 

27.8388 

9.1855 

830 

688900 

571787000 

28.8097 

9.3978 

776 

602176 

467288576 

27.8568 

9.1894 

83 

69056 

57385619 

28.8271 

9.4016 

777 

603729 

469097433 

27.8747 

9.^33 

832 

692224 

575930368 

28.8444 

9.4053 

778 

605284 

470910952 

27.8927 

9.1973 

833 

693889 

578009537 

28.8617 

9.4091 

779 

606341 

472729139 

27.9106 

9.2012 

834 

695556 

580093704 

28.8791 

9.4129 

780 

603400 

474552000 

27.9285 

9.2052 

835 

697225 

582182875 

28.8964 

9.4166 

781 

609961 

476379541 

27.9464 

9.2091 

836 

698896 

584277056 

28.9131 

9.4204 

782 

611524 

478211768 

27.9643 

9.2130 

837 

700569 

586376253 

28.9310 

9.4241 

783 

6(3089 

430048687 

27.9821 

9.2170 

838 

702244 

588480472 

28.9482 

9.4279 

784 

614656 

431890304 

28.0000 

9.2209 

839 

703921 

590589719 

28.9655 

9.4316 

785 

616225 

483736625 

28.0179 

9.2248 

840 

705600 

592704000 

28.9828 

9.4354 

786 

617796 

485587656 

28.0357 

9.2287 

841 

707281 

594823321 

29.0000 

9.4391 

787 

619369 

487443403 

28.0535 

9.2326 

842 

708964 

596947688 

29.0172 

9.4429 

788 

620944 

489303872 

28.0713 

9.2365 

843 

710649 

599077107 

29.0345 

9.4466 

789 

622521 

491169069 

28.0891 

9.2404 

844 

712336 

601211584 

29.0517 

9.4503 

790 

624100 

493039000 

28.1069 

9.2443 

845 

714025 

603351125 

29.0689 

9.4541 

791 

625631 

494913671 

28.1247 

9.2482 

846 

715716 

605495736 

29.0861 

9.4578 

792 

627264 

496793088 

28.1425 

9.2521 

847 

717409 

607645423 

29.1033 

9.4615 

793 

623349 

498677257 

28.1603 

9.2560 

848 

719104 

609800192 

29.1204 

9.4652 

794 

630436 

500566184 

28.1780 

9.2599 

849 

720801 

611960049 

29.1376 

9.4690 

795 

632025 

502459875 

28.1957 

9.2638 

850 

722500 

614125000 

29.1548 

9.4727 

796 

633616 

504358336 

28.2135 

9.2677 

851 

724201 

616295051 

29.1719 

9.4764 

797 

635209 

506261573 

28.23129.2716 

852 

725904 

618470208 

29.  1  890 

9.4801 

798 

636804 

508169592 

28.2489 

9.2754 

853 

727609 

620650477 

29.2062 

9.4838 

799 

638401 

510082399 

28.2666 

9.2793 

854 

729316 

622835864 

79.2233 

9.4875 

800 

640000 

512000000 

28.2843 

9.2832 

855 

731025 

625026375 

29.2404 

9.4912 

801 

641601 

513922401 

28.3019 

9.2870 

856 

732736 

627222016 

29.2575 

9.4949 

802 

643204 

515849608 

28.3196 

9.2909 

857 

734449 

629422793 

29.2746 

9.4986 

803 

644809 

517781627 

28.3373 

9.2948 

858 

736164 

631628712 

29.2916 

9.5023 

804 

646416 

519718464 

28.3549 

9.2986 

859 

737881 

633839779 

29.3087 

9.5060 

805 

648025 

521660125 

28.3725 

9.3025 

860 

739600 

636056000 

29.3258 

9.5097 

806 

649636 

523606616 

28.3901 

9.3063 

861 

741321 

638277381 

29.3428 

9.5134 

807 

651249 

525557943 

28.4077 

9.3102 

862 

743044 

640503928 

29.3598 

9.5171 

808 

652864 

527514112 

28.4253 

9.3140 

863 

744769 

642735647 

29.3769 

9.5207 

809 

654481 

529475129 

28.4429 

9.3179 

864 

746496 

644972544 

29.3939 

9.5244 

810 

656100 

531441000 

28.4605 

5.3217 

865 

748225 

647214625 

9.4109 

9.5231 

811 

657721 

533411731 

28.4781 

5.3255 

866 

49956 

49461896 

9.4279 

9.5317 

812 

659344 

35387328 

28.4956  9.3294 

867 

51689 

51714363  29.4449 

9.5354 

813 
814 

660969  37367797 
662596  539353144 

28.51329.3332 
28.5307  9.3370 

868  5342465397203229.4618 
8691  755  1  6  1  1  656234909!  29.4788 

9.5391 
9.5427 

102 


MATHEMATICAL   TABLES. 


No. 

870 

871 
872 
873 
874 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

No. 

~925 
926 
927 
928 
929 

Square 

Cube. 

Sq. 
Root. 

Cube 
Root. 

756900 
758641 
760384 
762129 
763876 

658503000 
660776311 
663054848 
665338617 
667627624 

29.4958 
29.5127 
29.5296 
29.5466 
29.5635 

9.5464 
9.5501 
9.5537 
9.5574 
9.5610 

855625 
857476 
859329 
861184 
863041 

791453125 
794022776 
796597983 
799178752 
801765089 

30.4138 
30.4302 
30.4467 
30.4631 
30.4795 

9.7435 
9.7470 
9.7505 
9.7540 
9.7575 

875 

876 
877 
878 
879 

765625 
767376 
769129 
770884 
772641 

669921875 
672221376 
674526133 
676836152 
679151439 

29.5804 
29.5973 
29.6142 
29.63  1  1 
29.6479 

9.5647 
9.5683 
9.5719 
9.5756 
9.5792 

930 
931 
932 
933 
934 

864900 
866761 
868624 
870489 
872356 

804357000 
806954491 
809557568 
812166237 
814780504 

30.4959 
30.5123 
30.5287 
30.5450 
30.5614 

9.76M 
9.7645 
9.7680 
9.7715 
9.7750 

880 
881 
882 
883 
884 

774400 
776161 
777924 
779689 
781456 

681472000 
683797841 
686128968 
688465387 
690807104 

29.6648 
29.6816 
29.6985 
29.7153 
29.7321 

9.5828 
9.5865 
9.5901 
9.5937 
9.5973 

935 
936 
937 
938 
939 

874225 
876096 
877969 
879844 
881721 

817400375 
820025856 
822656953 
825293672 
827936019 

30.5778 
30.5941 
30.6105 
30.6268 
30.6431 

9.7785 
9.7819 
9.7854 
9.7889 
9.7924 

885 
886 
887 
888 
889 

783225 
784996 
786769 
788544 
790321 

693154125 
695506456 
697864103 
700227072 
702595369 

29.7489 
29.7658 
29.7825 
29.7993 
29.8161 

9.6010 
9.6046 
9.6082 
9.6118 
9.6154 

940 
941 
942 
943 
944 

883600 
885481 
887364 
889249 
891136 

830584000 
833237621 
835896888 
838561807 
841232384 

30.6594 
30.6757 
30.6920 
30.7083 
30.7246 

9.7959 
9.7993 
9.8028 
9.8063 
9.8097 

890 
891 
892 
893 
694 

792100 
793881 
795664 
797449 
799236 

704969000 
707347971 
709732288 
712121957 
714516984 

29.8329 
29.8496 
29.8664 
29.8831 
29.8998 

9.6190 
9.6226 
9.6262 
9.6298 
9.6334 

945 
946 
947 
948 
949 

893025 
894916 
896809 
898704 
900601 

843908625 
846590536 
849278123 
851971392 
854670349 

30.7409 
30.7571 
30.7734 
30.7896 
30.8058 

9.8132 
9.8167 
9.8201 
9.8236 
9.8270 

895 
896 
897 
898 
899 

801025 
802816 
804609 
806404 
808201 

716917375 
719323136 
721734273 
724150792 
726572699 

29.9166 
29.9333 
29.9500 
29.9666 
29.9833 

9.6370 
9.6406 
9.6442 
9.6477 
9.6513 

950 
951 
952 
953 
954 

902500 
904401 
906304 
908209 
910116 

857375000 
860085351 
862801408 
865523177 
868250664 

30.8221 
30.8383 
30.8545 
30.8707 
30.8869 

9.8305 
9.8339 
9.8374 
9.8408 
9.8443 

900 
901 
902 
903 
904 

810000 
811801 
813604 
815409 
817216 

729000000 
731432701 
733870808 
736314327 
738763264 

30.0000 
30.0167 
30.0333 
30.0500 
30.0666 

9.6549 
9.6585 
9.6620 
9.6656 
9.6692 

955 
956 
957 
958 
959 

912025 
913936 
915849 
917764 
919681 

870983875 
873722816 
876467493 
879217912 
881974079 

30.9031 
30.9192 
30.9354 
30.9516 
30.9677 

9.8477 
9.8511 
9.8546 
9.8580 
9.8614 

905 
906 
907 
908 
909 

819025 
820836 
822649 
824464 
826281 

741217625 
743677416 
746142643 
748613312 
751089429 

30.0832 
30.0998 
30.1164 
30.1330 
30.1496 

9.6727 
9.6763 
9.6799 
9.6834 
9.6870 

960 
961 
962 
963 
964 

921600 
923521 
925444 
927369 
929296 

884736000 
887503681 
890277128 
893056347 
895841344 

30.9839 
31.0000 
31.0161 
31.0322 
31.0483 

9.8648 
9.8683 
9.8717 
9.8751 
9.8785 

910 
911 
912 
913 
914 

828100 
829921 
831744 
833569 
835396 

753571000 
75605803  1 
758550528 
761048497 
763551944 

30.1662 
30.1828 
30.1993 
30.2159 
30.2324 

9.6905 
9.6941 
9.6976 
9.7012 
9.7047 

965 
966 
967 
968 
969 

931225 
933156 
935089 
937024 
938961 

898632125 
901428696 
904231063 
907039232 
909853209 

3  1  .0644 
3  1  .0805 
3  1  .0966 
31.1127 
31.1288 

9.8819 
9.8854 
9.8888 
9.8922 
9.8956 

915 
916 
917 
918 
919 

837225 
839056 
840889 
842724 
844561 

766060875 
768575296 
771095213 
773620632 
776151559 

30.2490 
30.2655 
30.2820 
30.2985 
30.3150 

9.7082 
9.7118 
9.7153 
9.7188 
9.7224 

970 
971 
972 
973 

974 

940900 

942841 
944784 
946729 
948676 

912673000 
915498611 
918330048 
921167317 
924010424 

31.1448 
31.1609 
31.1769 
31.1929 
31.2090 

9.8990 
9.9024 
9.9058 
9.9092 
9.9126 

920 
921 
922 
923 
924 

846400 
848241 
850084 
851929 
853776 

778688000 
781229961 
783777448 
786330467 
7888890241 

30.3315 
30.3480 
30.3645 
30.3809 
30.3974 

9.7259 
9.7294 
9.7329 
9.7364 
9.7400 

975 
976 
977 
978 
979 

950625 
952576 
954529 
956484 
958441 

926859375 
929714176 
932574833 
935441352 
938313739 

31.2250 
31.2410 
31.2570 
31.2730 
31.2890 

9.9160 
99194 
9.9227 
9.9261 
9.9293 

SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.        103 


No. 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

No. 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

980 

960400 

941  192000 

31.3050 

9.9329 

1035 

1071225 

1108717875 

32.1714 

10.1153 

981 

962361 

944076141 

31.3209 

9.9363 

1036 

10732% 

1111934656 

32.1870 

10.1186 

982 

964324 

946966168 

31.3369 

9.93% 

1037 

1075369 

1115157653 

32.2025 

10.1218 

983 

966289 

949862087 

31.3528 

9.9430 

1038 

1077444 

1  1  18386872 

32.2180 

10.1251 

984 

968256 

952763904 

31.3688 

9.9464 

1039 

1079521 

1121622319 

32.2335 

10.1283 

985 

970225 

955671625 

31.3847 

9.9497 

1040 

1081600 

1124864000 

32.2490 

10.1316 

986 

9721% 

958585256 

31.4006 

9.9531 

1041 

1083681 

1128111921 

32.2645 

10  1348 

987 

974169 

%1  504803 

31.4166 

9.9565 

1042 

1085764 

1131366088 

32.2800 

10.1381 

988 

976144 

964430272 

31.4325 

9.9598 

1043 

1087849 

1134626507 

32.2955 

10.1413 

989 

978121 

%7361669 

31.4484 

9.%32 

1044 

1089936 

1137893184 

32.3110 

10.1446 

990 

980100 

970299000 

31.4643 

9.%66 

1045 

1092025 

1141166125 

32.3265 

10.1478 

991 

982081 

973242271 

31.4802 

9.%99 

1046 

1094116 

1144445336 

32.3419 

10.1510 

992 

984064 

976191488 

31.4960 

9.9733 

1047 

10%209 

1147730823 

32.3574 

10.1543 

993 

986049 

979146657 

31.5119 

9.9766 

1048 

1098304 

1151022592 

32.3728 

10.1575 

994 

988036 

982107784 

31.5278 

9.9800 

1049 

ir00401 

1154320649 

32.3883 

10.1607 

995 

990025 

985074875 

31.5436 

9.9833 

1050 

1  102500 

1157625000 

32.4037 

10.1640 

9% 

992016 

988047936 

31.5595 

9.9866 

1051 

1104601 

1160935651 

32.4191 

10.1672 

997 

994009 

991026973  31.5753 

9.9900 

1052 

1  106704 

1164252608 

32.4345 

10.1704 

998 

996004 

99401  1992 

31.5911 

9.9933 

1053 

1108809 

1167575877 

32.4500 

10.1736 

999 

998001 

997002999 

31.6070 

9.9%7 

1054 

1110916 

1170905464 

32.4654 

10.1769 

1000 

1000000 

1000000000 

31.6228 

10.0000 

1055 

1  1  13025 

1174241375 

32.4808 

10.1801 

1001 

1002001 

1003003001 

31.6386 

10.0033 

1056 

1115136 

1177583616 

32.4%2 

10.1833 

1002 

1004004 

1006012008 

31.6544 

10.0067 

1057 

1117249 

1180932193 

32.5115 

10.1865 

1003 

1006009 

1009027027 

31.6702 

10.0100 

1058 

1  1  19364 

1184287112 

32.5269 

10.1897 

1004 

1008016 

1012048064 

31.6860 

10.0133 

1059 

1121481 

1  187648379 

32.5423 

10.1929 

1005 

1010025 

1015075125 

31.7017 

10.0166 

1060 

1123600 

1191016000 

32.5576 

10.1%1 

1006 

1012036 

1018108216 

31.7175 

10.0200 

1061 

1125721 

1  194389981 

32.5730 

10.1993 

1007 

1014049 

1021147343 

31.7333 

10.0233 

1062 

1127844 

1  197770328 

32.5833 

10.2025 

1008 

1016064 

1024192512 

31.7490 

10.0266 

1063 

1129%9 

1201157047 

32.6036 

10.2057 

1009 

1018081 

1027243729 

31.7648 

10.0299 

1064 

11320% 

1204550144 

32.6190 

10.2089 

1010 

T020100 

1030301000 

31.7805 

10.0332 

1065 

1134225 

120794%25 

32  6343 

10.2121 

1011 

1022121 

1033364331 

31.7962 

10.0365 

1066 

1136356 

12113554% 

32.6497 

10.2153 

1012 

1024144 

1036433728 

31.8119 

10.0398 

1067 

1138489 

1214767763 

32.6650 

10.2185 

1013 

1026169 

1039509197 

31.8277 

10.0431 

1063 

1140624 

1218186432 

32.6803 

10.2217 

1014 

10281% 

1042590744 

31.8434 

10.0465 

1069 

1142761 

1221611509 

32.6956 

10.2249 

1015 

1030225 

1045678375 

31.8591 

10.0498 

1070 

1144900 

1225043000 

32.7109 

10.2281 

1016 

1032256 

10487720% 

31.8748 

10.0531 

1071 

1  147041 

1228480911 

32.7261 

10.2313 

1017 

1034289 

1051871913 

31.8904 

10.0563 

1072 

1149184 

1231925248 

32.7414 

10.2345 

1018 

1036324 

1054977832 

31.9061 

10.0596 

1073 

1151329 

1235376017 

32.7567 

10.2376 

1019 

1038361 

1058089859 

31.9218 

10.0629 

1074 

1153476 

1238833224 

32.7719 

10.2408 

1020 

1040400 

1061208000 

31.9374 

10.0662 

1075 

1155625 

1242296875 

32.7872 

10.2440 

1021 

1042441 

1064332261 

31.9531 

10.0695 

1076 

1157776 

1245766976 

32.8024 

10.2472 

1022 

1044484 

1067462648 

31.9687 

10.0728 

1077 

1159929 

1249243533 

32.8177 

10.2503 

1023 

1046529 

1070599167 

31.9844 

10.0761 

1078 

1162084 

1252726552 

32.8329 

10.2535 

1024 

1048576 

1073741824 

32.0000 

10.0794 

1079 

1  164241 

1256216039 

32.8481 

10.2567 

1025 

1050625 

1076890625 

32.0156 

10.0826 

1080 

1166400 

1259712000 

32.8634 

10.2599 

1026 

1052676 

1080045576 

32.0312 

10.0859 

1081 

1  168561 

1263214441 

32.8786 

10.2630 

1027 

1054729 

1083206683 

32.0468 

10.0892 

1032 

1170724 

1266723368 

32.8938 

10.2662 

1028 

1056784 

1086373952 

32.0624 

10.0925 

1033 

1172889 

1270238787 

32.9090 

10.2693 

1029 

1058841 

1089547389 

32.0780 

10.0957 

1084 

1175056 

1273760704 

32.9242 

10.2725 

1030 

1060900 

1092727000 

32.0936 

10.0990 

1035 

1177225 

1277289125 

32.9393 

10.2757 

1031 

1062%! 

1095912791 

32.1092 

10.1023 

1036 

11793% 

1280824056 

32.9545 

10.2788 

1032 

1065024 

1099104768 

32.1248 

10.1055 

1037 

1181569 

1284365503 

32.%97 

10.2820 

1033 

10670S9 

1  102302937 

32.1403 

10.1088 

1088 

1183744 

1287913472 

32.9848 

10.2851 

1034 

1069156 

1105507304 

32.1559 

10.1121 

1089 

11  8592  11  1291  467969 

33.0000 

10.2883 

104 


MATHEMATICAL   TABLES. 


No. 
T090 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

No. 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

1188100 

1295029000 

33.0151 

10.2914 

1145 

1311025 

1501123625 

33.8378 

10.4617 

1091 

1  190281 

12985%571 

33.0303i  10.2946 

1146 

1313316 

1505060136 

33.8526 

10.4647 

1092 
1093 

1192464 
1194649 

1302170688 
1305751357 

33.0454 
33.0606 

10.297/ 
10.3009 

1147 
1148 

1315609 
1317904 

1509003523 
1512953792 

33.8674 
33.882 

10.4678 
10.4708 

1094 

1196836 

1309338584 

33.0757 

10.3040 

1149 

132020 

1516910949 

33.8%9 

10.4739 

1095 

1199025 

1312932375 

33.0908 

10.307 

1150 

1322500 

1520875000 

33.9116 

10.4769 

10% 

1201216 

1316532736 

33.1059 

10.3103 

1151 

1324801 

1524845951 

33.9264 

10.4799 

1097 

1203409 

1320139673 

33.1210 

10.3134 

1152 

1327104 

1528823808 

33.9411 

10.4830 

1098 

1205604 

1323753192 

33.1361 

10.3165 

1153 

1329409 

1532808577 

33.9559 

10.4860 

1099 

1207801 

1327373299 

33.1512 

10.3197 

1154 

1331716 

1536800264 

33.9706 

10.4890 

1100 

1210000 

1331000000 

33.1662 

10.3228 

1155 

1334025 

1540798875 

33.9853 

10.4921 

1101 

1212201 

1334633301 

33.1813J  10.3259 

1156 

1336336 

1544804416 

34.0000 

10.4951 

1102 

1214404 

1338273208 

33.1964!  10.3290 

1157 

1338649 

1548816893 

34.0147 

10.4981 

1103 

1216609 

1341919727 

33  .21141  10.3322 

1158 

1340964 

1552836312 

34.0294 

10.5011 

1104 

1218816 

1345572864 

33.2264 

103353 

1159 

1343281 

1556862679 

34.0441 

10.5042 

1105 

1221025 

1349232625 

33.2415 

10.3384 

1160 

1345600 

1560896000 

34.0588 

10.5072 

1106 

1223236 

1352899016 

33.2566 

10.3415 

1161 

1347921 

1564936281 

34.0735 

10.5102 

1107 

1225449 

1356572043 

33.2716 

10.3447 

1162 

1350244 

1568933528 

34.0881 

10.5132 

1108 

1227664 

1360251712 

33.2866 

10.3478 

1163 

1352569 

1573037747 

34.1028 

1  0.5  162 

1109 

1229881 

1363938029 

33.3017 

10.3509 

1164 

13548% 

1577098944 

34.1174 

10.5192 

1110 

1232100 

1367631000 

33.3167 

10.3540 

1165 

1357225 

1581167125 

34.1321 

10.5223 

1111 

1234321 

1371330631 

33.3317 

10.3571 

1166 

1359556 

15852422% 

34.1467 

10.5253 

1112 

1236544 

1375036928 

33.3467 

10.3602 

1167 

1361889 

1589324463 

34.1614 

10.5283 

1113 

1238769 

1378749897 

33.3617 

10.3633 

1168 

1364224 

1593413632 

34.1760 

10.5313 

1114 

12409% 

1382469544 

33.3766 

10.3664 

1169 

1366561 

1597509809 

34.1906 

10.5343 

1115 

1243225 

1386195875 

33.3916 

10.3695 

1170 

1368900 

1601613000 

34.2053 

10.5373 

1116 

1245456 

13899288% 

33.4066 

10.3726 

1171 

1371241 

1605723211 

34.2199 

10.5403 

1117 

1247689 

1393668613 

33.4215 

103757 

1172 

1373584 

1609840448 

34.2345 

10.5433 

1118 

1249924 

1397415032 

33.4365 

10.3788 

1173 

1375929 

1613964717 

34.2491 

10.5463 

1119 

1252161 

1401168159 

33.4515 

10.3819 

1174 

1378276 

1618096024 

34.2637 

10.5493 

1120 

1254400 

1404928000 

33.4664 

10.3850 

1175 

1380625 

1622234375 

34.2783 

10.5523 

1121 

1256641 

408694561 

33.4813 

10.3881 

1176 

1382976 

1626379776 

34.2929 

10.5553 

1122 

1258884 

412467848 

33.4%3 

10.3912 

1177 

1385329 

1630532233 

34.3074 

10.5583 

1123 

1261  129 

416247867 

33.5112 

10.3943 

1178 

1387684 

1634691752 

34.3220 

10.5612 

1124 

1263376 

420034624 

33.5261 

10.3973 

1179 

1390041 

638858339 

34.3366 

10.5642 

1125 

1265625 

423828125 

33.5410 

10.4004 

1180 

1392400 

643032000 

34.3511 

10.5672 

1126 

1267876 

427628376 

33.5559 

10.4035 

1181 

1394761 

647212741 

34.3657 

10.5702 

1127 

1270129 

431435383 

33.5708 

10.4066 

1182 

1397124 

65140056834.3802 

10.5732 

1128 

1272384 

435249152 

33.5857 

10.4097 

1183 

1399489 

655595487 

34.3948 

0.5762 

1129 

1274641 

439069689 

33.6006 

10.4127 

1184 

1401856 

659797504 

34.4093 

0.5791 

1130 

1276900 

442897000 

33.6155 

10.4158 

1185 

1404225 

664006625 

34.4238 

0.5821 

\\3\ 

1279161 

446731091 

33.6303 

10.4189 

1186 

1406596 

668222856 

34.4384 

0.5851 

1132 

1281424 

450571968  33.6452 

10.4219 

1187 

140S%9 

672446203 

34.4529 

0.5881 

1133 

1283689 

4544  1%37|  33  .6601 

10.4250 

1158 

1411344 

676676672 

34.4674 

0.5910 

1134 

1285956 

458274104 

33.6749 

10.4281 

1189 

1413721 

680914269 

34.4819 

0.5940 

1135 

1288225 

462135375 

33.6898 

10.4311 

1190 

1416100 

685159000 

34.4964 

0.5970 

1136 

12904% 

466003456133.7046 

10.4342 

1191 

1418481 

689410871 

34.5109 

0.6000 

1137 

1292769 

469878353  33.7174 

10.4373 

1192 

1420864 

693669888 

34.5254 

0.6029 

1138 

1295044 

473760072  33.7342 

10.4404 

1193 

1423249 

697936057 

34.5398 

0.6059 

1139 

1297321 

477648619 

33.7491 

10.4434 

1194 

1425636 

702209384 

34.5543 

0.6088 

1140 

1299600 

481544000 

33.7639 

10.4464 

1195 

1428025 

706489875 

34,5688 

0.6118 

1141 

1301881 

485446221  33.7787 

10.4495 

11% 

1430416 

710777536 

34.5832 

0.6148 

1142 

1304164 

489355288  33.7935 

10  4525 

1197 

1432809 

715072373 

34.5977 

0.6177 

1143 

1306449 

493271207  33.8083 

10.4556 

1198 

1435204 

719374392 

34.6121 

0.6207 

1144 

1308736 

497193934338231 

10.4586 

1199 

1437601  '1723683599 

34.6266 

0.6236 

SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.       105 


No 
1200 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

No. 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

1440000 

1728000000 

34.6410 

10.6266 

1255 

1575025 

1976656375 

35.4260 

10.7865 

1201 

1442401 

1732323601 

34.6554 

10.6295 

1256 

1577536 

1981385216 

35.4401 

10.7894 

1202 

1444804 

1  736654408 

34.6699 

10.6325 

1257 

1580049 

1986121593 

35.4542 

10.7922 

1203 

1447209 

1740992427 

34.6843 

10.6354 

1258 

1582564 

1990865512 

35.4683 

10.7951 

1204 

1449616 

1745337664 

34.6987 

10.6384 

1259 

1585081 

1995616979 

35.4824 

10.7980 

1205 

1452025 

1749690125 

34.7131 

10.6413 

1260 

1587600 

2000376000 

35.4%5 

10.8008 

1206 

1454436 

1754049816 

34.7275 

10.6443 

1261 

1590121 

2005142581 

35.5106 

10.8037 

1207 

1456849 

1758416743 

34.7419 

10.6472 

1262 

1592644 

2009916728 

35.5246 

10.8065 

1208 

1459264 

1762790912 

34.7563 

1C.6501 

1263 

1595169 

2014698447 

35.5387 

10.8094 

1209 

1461631 

1767172329 

34.7707 

10.6530 

1264 

15976% 

2019487744 

35.5528 

10.8122 

1210 

1464100 

1771561000 

34.7851 

10.6560 

1265 

1600225 

202428.4625 

35.5668 

10.8151 

1211 

1  466521 

1775956931 

34.7994 

10.6590 

1266 

1602756 

2029089096 

35.5809 

10.8179 

1212 

1463944 

1780360128 

34.8138 

10.6619 

1267 

1605289 

2033901163 

35.5949 

10.8208 

1213 

1471369 

1784770597 

34.8281 

10.6648 

1268 

1607824 

2038720832 

35.6090 

10.8236 

1214 

1473796 

1789188344 

34.8425 

10.6678 

1269 

1610361 

2043548109 

35.6230 

10.8265 

1215 

1476225 

1793613375 

34.8569 

10.6707 

1270 

1612900 

2048383000 

35.6371 

10.8293 

1216 

1478656 

17980456% 

34.8712 

10.6736 

1271 

1615441 

2053225511 

35.6511 

10.8322 

1217 

1481089 

1802485313 

34.8855 

10.6765 

1272 

1617984 

2058075648 

35.6651 

10.8350 

1218 

1433524 

1806932232 

34.8999 

10.6795 

1273 

1620529 

2062933417 

35.6791 

10.8378 

1219 

1485%! 

1811386459 

34.9142 

10.6324 

1274 

1623076 

2067798824 

35.6931 

10.8407 

1220 

1438400 

1815848000 

34.9285 

10.6853 

1275 

1625625 

2072671875 

35.7071 

10.8435 

1221 

1490841 

1820316861 

34.9428 

10.6882 

1276 

1628176 

2077552576 

35.7211 

10.8463 

1222 

1493284 

1824793048 

34.9571 

10.691  1 

1277 

1630729 

2082440933 

35.7351 

10.8492 

1223 

1495729 

1829276567 

34.9714 

10.6940 

1278 

1633284 

2087336952 

35.7491 

10.8520 

1224 

1498176 

1833767424 

34.9357 

10.6970 

1279 

1635841 

2092240639 

35.7631 

10.8548 

1225 

1500625 

1838265625 

35.0000 

10.6999 

1280 

1638400 

2097152000 

35.7771 

10.857; 

1226 

1503076 

1842771176 

35.0143 

10.7028 

1281 

1640%! 

2102071041 

35.791  1 

10.8605 

1227 

1505529 

1847284033 

35.0286 

10.7057 

1282 

1643524 

2106997768 

35.8050 

10.8633 

1223 

1507984 

1851804352 

35.0428 

10.7086 

1283 

1646089 

2111932187 

35.8190 

10.8661 

1229 

1510441 

1856331989 

35.0571 

10.7115 

1284 

1648656 

21  16874304 

35.8329 

10.8690 

1230 

1512900 

1860867000 

35.0714 

10.7144 

1285 

1651225 

2121824125 

35.8469 

10.8718 

1231 

1515361 

1865409391 

35.0856 

10.7173 

1286 

1653796 

2126781656 

35.8608 

10.8746 

1232 

1517824 

1869959163 

35.0999 

10.7202 

1287 

1656369 

2131746903 

35.8748 

10.8774 

1233 

1520239 

1874516337 

35.1141 

10.7231 

1238 

1658944 

2136719872 

35.8887 

10.8802 

1234 

1522756 

1879080904 

35.1283 

10.7260 

1289 

1661521 

2141700569 

35.9026 

10.8831 

1235 

1525225 

1833652875 

35.1426 

10.7289 

1290 

1664100 

2146689000 

35.9166 

10.8859 

1236 

1527696 

1838232256 

35.1568 

10.7318 

1291 

1666681 

2151685171 

35.9305 

10.8887 

1237 

1530169 

1892819053 

35.1710 

10.7347 

1292 

1669264 

2156689088 

35.9444 

10.8915 

1233 

1532644 

1897413272 

35.1852 

10.7376 

1293 

1671849 

2161700757 

35.9583 

10.8943 

1239 

1535121 

1902014919 

35.1994 

10.7405 

1294 

1674436 

2166720184 

35.9722 

10.8971 

1240 

1537600 

1906624000 

35.2136 

10.7434 

1295 

1677025 

2171747375 

35.9fBl 

10.8959 

1241 

1540081 

1911240521 

35.2278 

10.7463 

1296 

167%16 

2176782336 

36.0000 

10.9027 

1242 

1542564 

1915864438 

35.2420 

10.7491 

1297 

1682209 

2181825073 

36.0139 

10.9055 

1243 

1545049 

1920495907 

35.2562 

10.7520 

1298 

1684804 

2186875592 

36.0278 

10.9083 

1244 

1547536 

1925134784 

35.2704 

10.7549 

1299 

1687401 

2191933899 

36.0416 

10.9111 

1245 

1550025 

1929781125 

35.2846 

10.7578 

1300 

1690000 

2197000000 

36.0555 

10.9139 

1246 

1552516 

1934434936 

35.2987 

10.7607 

1301 

1692601 

2^02073901 

36.0694 

10.9167 

1247 

1555005 

1939096223 

35.3129 

10.7635 

1302 

1695204 

2207155608 

36.0832 

10.9195 

1243 

1557504 

1  943764992 

35.3270 

10.7664 

1303 

1697809 

2212245127 

36.0971 

10.9223 

1249 

1560001 

1948441249 

35.3412 

10.7693 

1304 

1700416 

2217342464 

36.1109 

10.9251 

1250 

1562509 

1953125000 

35.3553 

10.7722 

1305 

1703025 

2222447625 

36.1248 

10.9279 

1251 

1555011 

1957816251 

35.3695 

10.7750 

1306 

1705636 

2227560616 

36.1386 

10.9307 

1252 

1567504 

1962515008 

35.3836 

10.7779 

1307 

1708249 

2232681443 

36.1525 

10.9335 

1253 

1570009 

1967221277 

35.3977 

10.7808 

1308 

1710864 

22378101  12 

36.1663 

10.9363 

1254 

1572516 

197193506435.4119 

10.7837 

1309 

1713481 

2242946629 

36.1801 

10.9391 

106 


MATHEMATICAL  TABLES. 


No. 

Square. 

Cube. 

Sq. 
Root. 

Cube 
.Root. 

No. 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

T310 

1716100 

2248091000 

36.1939 

10.9418 

1365 

1863225 

2543302125 

36.9459 

1  1  .0929 

1311 

1718721 

2253243231 

36.2077 

10.9446 

1366 

1865956 

25488958% 

36.9594 

1  1  .0956 

1312 

1721344 

2258403328 

36.2215 

10.9474 

1367 

1868689 

2554497863 

36.9730 

1  1  .0983 

1313 

1723969 

2263571297 

36.2353 

10.9502 

1368 

1871424 

2560108032 

36.9865 

11.1010 

1314 

17265% 

2268747144 

36.2491 

10.9530 

1369 

1874161 

2565726409 

37.0000 

11.1037 

1315 

1729225 

2273930875 

36.2629 

10.9557 

1370 

1876900 

2571353000 

37.0135 

11.1064 

1316 

1731856 

22791224% 

36.2767 

10.9585 

1371 

1879641 

257698781  1 

37.0270 

11.1091 

1317 

1734489 

2284322013 

36.2905 

10.%13 

1372 

1882384 

2582630848 

37.0405 

11.1118 

1318 

1737124 

2289529432 

36.3043 

10.9640 

1373 

1885129 

2588282117 

37.0540 

11.1145 

1319 

1739761 

2294744759 

36.3180 

10.9668 

1374 

1887876 

2593941624 

37.0675 

11.1172 

1320 

1742400 

2299968000 

36.3318 

10.%% 

1375 

1890625 

2599609375 

37.0810 

11.1199 

1321 

1745041 

2305199161 

36.3456 

10.9724 

1376 

1893376 

2605285376 

37.0945 

11.1226 

1322 

1747684 

2310438248 

36.3593 

10.9752 

1377 

18%129 

261096%33 

37.1080 

11.1253 

1323 

1750329 

2315685267 

36.3731 

10.9779 

1378 

1898884 

2616662152 

37.1214 

11.1280 

1324 

1752976 

2320940224 

36.3868 

10.9807 

1379 

1901641 

2622362939 

37.1349 

11.1307 

1325 

1755625 

2326203125 

36.4005 

10.9834 

1380 

1904400 

2628072000 

37.1484 

11.1334 

1326 

1758276 

2331473976 

36.4143 

10.9862 

1381 

1907161 

2633789341 

37.1618 

11.1361 

1327 

1760929 

2336752783 

36.4280 

10.9890 

1382 

1909924 

2639514968 

37.1753 

11.1387 

1328 

1763584 

2342039552 

36.4417 

10.9917 

1383 

1912689 

2645248887 

37.1887 

11.1414 

1329 

1766241 

2347334289 

36.4555 

10.9945 

1384 

1915456 

2650991104 

37.2021 

11.1441 

1330 

1768900 

2352637000 

36.4692 

10.9972 

1385 

1918225 

2656741625 

37.2156 

11.1468 

1331 

1771561 

2357947691 

36.4829 

11.0000 

1386 

1920996 

2662500456 

37.2290 

11.1495 

1332 

1774224 

2363266368 

36.4966 

1  1  .0028 

1387 

1923769 

2668267603 

37.2424 

11.1522 

1333 

1776889 

2368593037 

36.5103 

11.0055 

1388 

1926544 

2674043072 

37.2559 

11.1548 

1334 

1779556 

2373927704 

36.5240 

11.0083 

1389 

1929321 

2679826869 

37.2693 

11.1575 

1335 

1782225 

2379270375 

36.5377 

11.0110 

1390 

1932100 

2685619000 

37.2827 

11.1602 

1336 

1784896 

2384621056 

36.5513 

11.0138 

1391 

1934881 

2691419471 

37.2961 

11.1629 

1337 

1787569 

2389979753 

36.5650 

11.0165 

1392 

1937664 

2697228288 

37.3095 

11.1655 

1338 

1790244 

2395346472 

36.5787 

11.0193 

1393 

1940449 

2703045457 

37.3229 

11.1682 

1339 

1792921 

2400721219 

36.5923 

11.0220 

1394 

1943236 

2708870984 

37.3363 

11.1709 

1340 

1795600 

2406104000 

36.6060 

11.0247 

1395 

1946025 

2714704875 

37.3497 

11.1736 

1341 

1798281 

2411494821 

36.6197 

1  1  .0275 

1396 

1948816 

2720547136 

37.3631 

11.1762 

1342 

1800964 

2416893688 

36.6333 

1  1  .0302 

1397 

1951609 

2726397773 

37.3765 

11.1789 

1343 

1803649 

2422300607 

36.6469 

1.0330 

1398 

1954404 

2732256792 

37.3898 

11.1816 

1344 

1806336 

2427715584 

36.6606 

1  .0357 

1399 

1957201 

2738124199 

37.4032 

11.1842 

1345 

1809025 

2433138625 

36.6742 

.0384 

1400 

1960000 

2744000000 

37.4166 

11.1869 

1346 

1811716 

2438569736 

36.6879 

.0412 

1401 

1962801 

2749884201 

37.4299 

11.1896 

1347 

1814409 

2444008923 

36.7015 

.0439 

1402 

1%5604 

2755776808 

37.4433 

11.1922 

1348 

1817104 

2449456192 

36.7151 

.0466 

1403 

1%8409 

2761677827 

37.4566 

11.1949 

1349 

1819801 

2454911549 

36.7287 

.0494 

1404 

1971216 

2767587264 

37.4700 

11.1975 

1350 

1822WO 

2460375000 

36.7423 

.0521 

1405 

1974025 

2773505125 

37.4833 

11.2002 

1351 

1825201 

2465846551 

36.7560 

1  .0548 

1406 

1976836 

2779431416 

37,4967 

1  1  .2028 

1352 

1827904 

2471326208 

36.76% 

1  .0575 

1407 

1979649 

2785366143 

37.5100 

1  1  .2055 

1353 

1830609 

2476813977 

36.7831 

.0603 

1408 

1982464 

2791309312 

37.5233 

1  1  .2082 

1354 

1833316 

2482309864 

36.7967 

1.0630 

1409 

1985281 

2797260929 

37.5366 

11.2108 

1355 

1836025 

2487813875 

36.8103 

.0657 

1410 

1988100 

2803221000 

37.5500 

11.2135 

1356 

1838736 

2493326016 

36.8239 

1.0684 

1411 

1990921 

2809189531 

37.5633 

11.2161 

1357 

1841449 

2498846293 

36.8375 

1.0712 

1412 

1993744 

2815166528 

37.5766 

11.  2  188 

1358 

1844164 

2504374712 

36.8511 

1.0739 

1413 

19%569 

2821151997 

37.5699 

11.2214 

1359 

1846881 

2509911279 

36.8646 

1  .0766 

1414 

1999396 

2827145944 

37.6032 

1  1  .2240 

1360 

1849600 

2515456000 

36.8782 

1  .0793 

1415 

2002225 

2833148375 

37  6165 

11.2267 

1361 

1852321 

2521008881 

36.8917 

1  .0820 

1416 

2005056 

28391592% 

37.6298 

11.2293 

1362 

1855044 

2526569928 

36.9053 

1  .0847 

1417 

2007889 

2845178713 

37.6431 

11.2320 

1363 

1857769 

2532139147 

36.9188  11.0875 

1418 

2010724 

2851206632 

37.6563 

1  1  2346 

1364  18604% 

2537716544 

36.9324  1  1  .0902 

1419 

2013561 

2857243059 

37.66% 

1  1  2373 

SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.       107 


No. 

Square. 

Cube. 

Sq.  | 
Root. 

Cube 
Root. 

No. 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

1420 
1421 
1422 
1423 
1424 

2016400 
2019241 
2022084 
2024929 
2027776 

2863288000 
2869341461 
2875403448 
2881473967 
2887553024 

37.6829 
37.6962 
37.7094 
37.7227 
37.7359 

1  1  .2399 
1  1  .2425 
1  1  .2452 
11.2478 
11.2505 

1475 
1476 
1477 
1478 
1479 

2175625 
2178576 
2181529 
2184484 
2187441 

3209046875 
3215578176 
3222118333 
3228667352 
3235225239 

38.4057 

38  4187 
38.4318 
38.4448 
38.4578 

1  1  .3832 
1  1  .3858 
1  1  .3883 
11.3909 
11.3935 

1425 
1426 
1427 
1428 
1429 

2030625 
2033476 
2036329 
2039184 
2042041 

2893640625 
2899736776 
2905841483 
2911954752 
2918076589 

37.7492 
37.7624 
37.7757 
37.7889 
37.8021 

11.2531 
11.2557 
1  1  .2583 
11.2610 
11.2636 

1480 
1481 
1482 
1483 
1484 

2190400 
2193361 
21%324 
2199289 
2202256 

3241792000 
3248367641 
3254952168 
3261545587 
3268147904 

38.4708 
38.4838 
38.4968 
38.5097 
38.5227 

11.3960 
11.3986 
11.4012 
11.4037 
11.4063 

1430 
1431 
1432 
1433 
1434 

2044900 
2047761 
2050624 
2053489 
2056356 

2924207000 
2930345991 
2936493568 
2942649737 
2948814504 

37.8153 
37.8286 
37.8418 
37.8550 
37.8682 

11.2662 
11.2689 
11.2715 
11.2741 
11.2767 

1485 
1486 
1487 
1488 
1489 

2205225 
2208196 
2211169 
2214144 
2217121 

3274759125 
3281379256 
3288008303 
3294646272 
3301293169 

38.5357 
38.5487 
38.5616 
38.5746 
38.5876 

11.4089 
11.4114 
11  4140 
11.4165 
11.4191 

1435 
1436 
1437 
1438 
1439 

2059225 
20620% 
2064969 
2067844 
2070721 

2954987875 
2961169856 
2967360453 
2973559672 
29797675  19 

37.8814 
37.8946 
37.9078 
37.9210 
37.9342 

11.2793 
11.2820 
1  1  .2846 
11.2872 
1  1  .2898 

1490 
1491 
1492 
1493 
1494 

2220100 
2223081 
2226064 
2229049 
2232036 

3307949000 
3314613771 
3321287488 
3327970157 
3334661784 

386005 
38.6135 
386264 
38.6394 
38.6523 

11.4216 
11.4242 
1  1  .4268 
1  1  .4293 
11.4319 

1440 
1441 
1442 
1443 
1444 

2073600 
2076481 
2079364 
2032249 
2085136 

2985984000 
2992209121 
2998442888 
3004685307 
3010936384 

37.9473 
37.9605 
37.9737 
37.9868 
38.0000 

1  1  .2924 
11.2950 
1  1  .2977 
1  1  .3003 
11.3029 

1495 
1496 
1497 
1493 
1499 

2235025 
2238016 
2241009 
2244004 
2247001 

3341362375 
3348071936 
3354790473 
3361517992 
3368254499 

38.6652 
38.6782 
38.691  1 
38.7040 
38.7169 

1  1  .4344 
1  1  .4370 
11.4395 
11.4421 
11.4446 

1445 
1446 
1447 
1448 
1449 

2088025 
2090916 
2093809 
2096704 
2099601 

3017196125 
3023464536 
3029741623 
3036027392 
3042321849 

38.0132 
38.0263 
38.0395 
38  0526 
38.0657 

1  1  .3055 
11.3081 
11.3107 
11.3133 
11.3159 

1500 
1501 
1502 
1503 
1504 

2250000 
2253001 
2256004 
2259009 
2262016 

3375000000 
3381754501 
3388518008 
3395290527 
3402072064 

38.7298 
38.7427 
38.7556 
38.7685 
38.7814 

11.4471 
1  1  .4497 
11.4522 
1  1  .4548 
11.4573 

1450 
1451 
1452 
1453 
1454 

2102500 
2105401 
2108304 
2111209 
2114116 

3048625000 
3054936851 
3061257408 
3067586677 
3073924664 

38.0789 
38.0920 
38.1051 
38.1182 
38.1314 

11.3185 
11.3211 
1  1  .3237 
1  1  .3263 
11.3289 

1505 
1506 
1507 
1508 
1509 

2265025 
2268036 
2271049 
2274064 
2277081 

3408862625 
3415662216 
3422470843 
3429288512 
3436115229 

38.7943 
38.8072 
38  8201 
38.8330 
38.8458 

11.4598 
11.4624 
11.4649 
11.4675 
11.4700 

1455 
1456 
1457 
1458 
1459 

2117025 
2119936 
2122849 
2125764 
2128681 

3080271375 
3086626816 
3092990993 
3099363912 
3105745579 

38.1445 
38.1576 
38.1707 
38.1838 
38.1969 

11.3315 
11.3341 
1  1  .3367 
11.3393 
11.3419 

1510 
1511 
1512 
1513 
1514 

2280100 
2283121 
2286144 
2289169 
22921% 

3442951000 
3449795831 
3456649728 
3463512697 
3470384744 

38.8587 
38.8716 
38.8844 
38.8973 
38.9102 

11.4725 
11.4751 
11  ,4776 
11.4801 
11.4826 

1460 
1461 
1462 
1463 
1464 

2131600 
2134521 
2137444 
2140369 
2143296 

3112136000 
3118535181 
3124943128 
3131359847 
3137785344 

38.2099 
38.2230 
38.2361 
38.2492 
38.2623 

11.3445 
11.3471 
11.34% 
11.3522 
11.3548 

1515 

1516 
1517 
1518 
1519 

2295225 
2298256 
2301289 
2304324 
2307361 

3477265875 
34841560% 
3491055413 
3497%3832 
3504881359 

38.9230 
38.9358 
38.9487 
38.%15 
38.9744 

11.4852 
11.4877 
11.4902 
11.4927 
11.4953 

1465 
1466 
1467 
1468 
1469 

2146225 
2149156 
2152089 
2155024 
2157%1 

3144219625 
3150662696 
3157114563 
3163575232 
3170044709 

38.2753 
38.2884 
38.3014 
38.3145 
38.3275 

11.3574 
11.3600 
11.3626 
1  1  .3652 
11.3677 

1520 
1521 
1522 
1523 
1524 

2310400 
2313441 
2316484 
2319529 
2322576 

3511808000 
3518743761 
3525688648 
3532642667 
3539605824 

38.9872 
39.0000 
39.0128 
39.0256 
39.0384 

11.4978 
11.5003 
11.5028 
11.5054 
11.5079 

1470 
1471 
M72 
1473 
1474 

2160900 
2163841 
2166784 
2169729 
2172676 

3176523000 
3183010111 
3189506048 
3196010817 
3202524424 

38.3406 
38.3536 
38.3667 
38.3797 
38  3927 

11.3703 
1  1  .3729 
1  1  .3755 
1  1  3780 
11.3806 

1525 
1526 
1527 
4528 
1529 

2325625 
2328676 
2331729 
2334784 
2337841 

3546578125 
3553559576 
3560550183 
3567549952 
3574558889 

39.0512 
39.0640 
39.0768 
39.08% 
39.1024 

11.5104 
11.5129 
11.5154 
11.5179 
11.5204 

108 


MATHEMATICAL  TABLES, 


No. 

1530 
1531 
1532 
1533 
1534 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

No. 

"7565 
1566 
1567 
1568 
1569 

Square. 

Cube. 

Sq. 
Root. 

Cube 
Root. 

2340900 
2343961 
2347024 
2350089 
2353156 

3581577000 
3588604291 
3595640768 
3602686437 
3609741304 

39.1152 
39.1280 
39.1408 
39.1535 
39.1663 

1  1  .5230 
11.5255 
11.5280 
11.5305 
11.5330 

2449225 
2452356 
2455489 
2458624 
2461761 

3833037125 
38403894% 
3847751263 
3855123432 
3862503009 

39.5601 
39.5727 
39.5854 
39.5980 
39.6106 

11.6102 
11.6126 
11.6151 
11.6176 
11.6200 

1535 
1536 
1537 
1538 
1539 

2356225 
23592% 
2362369 
2365444 
2368521 

3616805375 
3623878656 
3630%  11  53 
3638052872 
3645153819 

39.1791 
39.1918 
39.2046 
39.2173 
39.2301 

11.5355 
1  1  .5380 
1  1  .5405 
1  1  .5430 
1  1  .5455 

1570 
1571 
1572 
1573 
1574 

2464900 
2468041 
2471184 
2474329 
2477476 

3869893000 
3877292411 
3884701248 
3892119517 
3899547224 

39.6232 
39.6358 
39.6485 
39.661  1 
39.6737 

11.6225 
1  1  .6250 
1  1  .6274 
1  1  .6299 
11.6324 

1540 
1541 
1542 
1543 
1544 

2371600 
2374681 
2377764 
2380849 
2383936 

3652264000 
3659383421 
3666512088 
3673650007 
3680797184 

39.2428 
39.2556 
39.2683 
39.2810 
39.2938 

1  1  .5480 
11.5505 
11.5530 
11.5555 
11.5580 

1575 
1576 
1577 
1578 
1579 

2480625 
2483776 
2486929 
2490084 
2493241 

3906984375 
3914430976 
3921887033 
3929352552 
3936827539 

39.6863 
39.6989 
39.7115 
39.7240 
39.7366 

1  1  .6348 
1  1  .6373 
1  1  .6398 
1  1  .6422 
1  1  .6447 

1545 
1546 
1547 
1548 
154Q 

2387025 
2390116 
2393209 
2396304 
2399401 

3687953625 
3695119336 
3702294323 
3709478592 
3716672149 

39.3065 
393192 
39.3319 
39.3446 
39.3573 

1  1  .5605 
1  1  .5630 
1  1  .5655 
11.5680 
11.5705 

1580 
1581 
1582 
1583 
1584 

2496400 
2499561 
2502724 
2505889 
2509056 

3944312000 
3951805941 
3959309368 
3966822287 
3974344704 

39.7492 
39.7618 
39.7744 
39.7869 
39.7995 

11.6471 
11.64% 
11.6520 
1  1  .6545 
1  1  .6570 

1550 
1551 
1552 
1553 
1554 

2402500 
2405601 
2408704 
2411809 
2414916 

3723875000 
3731087151 
3738308608 
3745539377 
3752779464 

39.3700 
39.3827 
39.3954 
39.4081 
39.4208 

1  1  .5729 
11.5754 
1  1  .5779 
1  1  .5804 
1  1  .5829 

1585 
1586 
1587 
1588 
1589 

2512225 
25153% 
2518569 
2521744 
2524921 

3981876625 
3989418056 
3996%9003 
4004529472 
4012099469 

39.8121 
39.8246 
39.8372 
39.8497 
39.8623 

1  1  .6594 
11.6619 
1  1  .6643 
11.6668 
1  1  .6692 

1555 
1556 
1557 
1558 
1559 

2418025 
2421136 
2424249 
2427364 
2430481 

3760028875 
3767287616 
3774555693 
3781833112 
37891  19879 

39.4335 
39.4462 
39.4588 
39.4715 
39.4842 

11.5854 
1  1  .5879 
1  1  .5903 
1  1  .5928 
1  1  .5953 

1590 
1591 
1592 
1593 
1594 

2528100 
2531281 
2534464 
2537649 
2540836 

401%79000 
4027268071 
4034866688 
4042474857 
4050092584 

39.8748 
39.8873 
39.8999 
39.9124 
39.9249 

11.6717 
1  1  .6741 
1  1  .6765 
1  1  .6790 
11.6814 

1560 
1561 
1562 
1563 
1564 

2433600 
2436721 
2439844 
2442969 
24460% 

3796416000 
3803721481 
381  1036328 
3818360547 
3825694144 

39.4968 
39.5095 
39.5221 
39.5348 
39.5474 

1  1  .5978 
11.6003 
11.6027 
1  1  .6052 
11.6077 

1595 
15% 
1597 
1598 
1599 

2544025 
2547216 
2550409 
2553604 
2556801 

4057719875 
4065356736 
4073003173 
4080659192 
4088324799 

39.9375 
39.9500 
39.%25 
39.9750 
39.9875 

1  1  .6839 
1  1  .6863 
11  6888 
11.6912 
11.6936 

1600 

2560000 

4096000000 

40.0000 

11.6961 

SQUARES  AND  CUBES  OF  DECIMALS. 


No. 

Square 

Cube. 

No. 

Square 

Cube. 

No. 

Square. 

'  Cube. 

\2 

.01 
.04 

.001 
.008 

.01 
.02 

.0001 
.0004 

.000  001 
.000  008 

.001 
.002 

.00  00  01 
.00  00  04 

.000  000  001 
.000  000  008 

.09 

.027 

.03 

.0009 

.000  027 

.003 

.00  00  09 

.000  000  027 

*4 

.16 

.064 

.04 

.0016 

.000  064 

.004 

.00  00  16 

,000  000  064 

.5 

.25 

.125 

.05 

.0025 

.000  125 

.005 

.00  00  25 

.000  000  125 

6 

.36 

.216 

.06 

.0036 

.000  216 

.006 

.00  00  36 

.000  000  216 

.7 

.49 

.343 

.07 

.0049 

.000  343 

.007 

.00  00  49 

.000  000  343 

8 

.64 

.512 

.08 

.0064 

.000  512 

.008 

.00  00  64 

.000  000  512 

.9 

.81 

.729 

.09 

.0081 

.000  729 

.009 

.00  00  81 

.000  000  729 

1  0 

1  00 

1.000 

.10 

.0100 

.001  000 

.010 

.00  01  00 

.000  001  000 

1.44 

1.728 

.12 

.0144 

.001  728 

.012 

.00  01  44 

.000  001  728 

Note  that  the  square  has  twice  as  many  decimal  places,  and  the  cube 
.three times  as  many  decimal  places,  as  the  root. 


FIFTH  ROOTS   AND   FIFTH   POWERS, 


109 


FIFTH  ROOTS  AND  FIFTH  POWERS. 

(Abridged  from  TRAUTWINB.) 


*i 
&& 

Power. 

o  3 

£« 

Power. 

(H   . 

o  -^ 

ll 

Power. 

S<i 

ll 

Power. 

li 

Itf 

Power. 

.10 

.000010 

3.7 

693.440 

9.8 

90392 

21.8 

4923597 

40 

102400000 

.15 

.000075 

3.8 

792.352 

9.9 

95099 

22.0 

5153632 

41 

115856201 

.20 

.000320 

3.9 

902.242 

10.0 

100000 

22.2 

5392186 

42 

130691232 

.25 

.000977 

4.0 

1024.00 

10.2 

110408 

22.4 

5639493 

43 

147008443 

.30 

.002430 

4.1 

1158.56 

10.4 

121665 

22.6 

5895793 

44 

164916224 

.35 

.005252 

4.2 

1306.91 

10.6 

133823 

22.8 

6161327 

45 

184528125 

.40 

.010240 

4.3 

1470.08 

10.8 

146933 

23.0 

6436343 

46 

205962976 

.45 

.018453 

4.4 

1649.16 

11.0 

161051 

23.2 

6721093 

47 

229345007 

.50 

.031250 

4.5 

1845.28 

11.2 

176234 

23.4 

7015834 

48 

254803968 

.55 

.050328 

4.6 

2059.63 

11.4 

192541 

23.6 

7320825 

49 

282475249 

.60 

.077760 

4.7 

2293.45 

11.6 

210034 

23.8 

7636332 

50 

312500000 

.65 

.116029 

4.8 

2548.04 

11.8 

228776 

24.0 

7962624 

51 

345025251 

.70 

.168070 

49 

2824.75 

12.0 

248832 

24.2 

8299976 

52 

380204032 

.75 

.237305 

5.0 

3125.00 

12.2 

270271 

24.4 

8648666 

53 

418195493 

.80 

.327680 

5.1 

3450.25 

12.4 

293163 

24.6 

9008978 

54 

459165024 

.85 

.443705 

5.2 

3802.04 

12.6 

317580 

24.8 

9381200 

55 

503284375 

.90 

.590490 

5.3 

4181.95 

12.8 

343597 

25.0 

9765625 

56 

550731776 

.95 

.773781 

5.4 

4591  65 

13.0 

371293 

25.2 

10162550 

57 

601692057 

.00 

1.00000 

5.5 

5032.84 

13.2 

400746 

25.4 

10572278 

58 

656356768 

.05 

1.27628 

5.6 

5507.32 

13.4 

432040 

25.6 

10995116 

59 

714924299 

.10 

1.61051 

5.7 

6016.92 

13.6 

465259 

25.8 

11431377 

60 

777600000 

.15 

2.01135 

5.8 

6563.57 

13.8 

500490 

26.0 

11881376 

61 

844596301 

.20 

2.48832 

5.9 

7149.24 

14.0 

537824 

26.2 

12345437 

62 

916132832 

.25 

3.05176 

6.0 

7776.00 

14.2 

577353 

26.4 

12823886 

63 

992436543 

.30 

3.71293 

6.1 

8445.96 

14.4 

619174 

26.6 

13317055 

64 

1073741824 

.35 

4.48403 

6.2 

9161.33 

14.6 

663383 

26.8 

13825281 

65 

1160290625 

.40 

5.37824 

6.3 

9924.37 

14.8 

710082 

27.0 

14348907 

66 

1252332576 

.45 

6.40973 

6.4 

10737 

15.0 

759375 

27.2 

14888280 

67 

1350125107 

.50 

7.59375 

6.5 

11603 

15.2 

811368 

27.4 

15443752 

68 

1453933568 

.55 

8.94661 

6.6 

12523 

15.4 

866171 

27.6 

16015681 

69 

1564031349 

.60 

10.4858 

6.7 

13501 

15.6 

923896 

27.8 

1  6604430 

70 

1680700000 

.65 

12.2298 

6.8 

14539 

15.8 

984658 

28.0 

17210368 

71 

1804229351 

.70 

14.1986 

6.9 

15640 

16.0 

1048576 

28.2 

17833868 

72 

1934917632 

.75 

16.4131 

7.0 

16807 

16.2 

1115771 

28.4 

18475309 

73 

2073071593 

.80 

18.8957 

7.1 

18042 

16.4 

1186367 

28.6 

19135075 

74 

2219006624 

.85 

21.6700 

7.2 

19349 

16.6 

1260493 

28.8 

19813557 

75 

2373046875 

.90 

24.7610 

7.3 

20731 

16.8 

1338278 

29.0 

20511149 

76 

2535525376 

.95 

28.1951 

7.4 

22190 

17.0 

1419857 

29.2 

21228253 

77 

2706784157 

2.00 

32.0000 

7.5 

23730 

17.2 

1  505366 

29.4 

21965275 

78 

2887174368 

2.05 

36.2051 

7.6 

25355 

17.4 

1594947 

29.6 

22722628 

79 

3077056399 

2.10 

40.8410 

7.7 

27068 

17.6 

1688742 

298 

23500728 

80 

3276800000 

2.15 

45.9401 

7.8 

28872 

17.8 

1  786899 

30.0 

24300000 

81 

3486784401 

2.20 

51.5363 

7.9 

30771 

18.0 

1889568 

30.5 

26393634 

82 

3707398432 

2.25 

57.6650 

8.0 

32768 

18.2 

1996903 

31.0 

28629151 

83 

3939040643 

2.30 

64.3634 

8.1 

34868 

18.4 

2109061 

31.5 

31013642 

84 

4182119424 

2.35 

71.6703 

8.2 

37074 

18.6 

2226203 

32.0 

33554432 

85 

4437053125 

2.40 

79.6262 

8.3 

39390 

18.8 

2348493 

32.5 

36259082 

86 

4704270176 

2.45 

88.2735 

8.4 

41821 

19.0 

2476099 

33.0 

39135393 

87 

4984209207 

2.50 

97.6562 

8.5 

44371 

19.2 

2609193 

33.5 

42191410 

88 

5277319168 

2.55 

107.820 

8.6 

47043 

19.4 

2747949 

34.0 

45435424 

89 

5584059449 

2.60 

118.814 

8.7 

49842 

19.6 

2892547 

34.5 

48875980 

90 

5904900000 

2.70 

143.489 

8.8 

52773 

19.8 

3043168 

35.0 

52521875 

91 

6240321451 

2.80 

172.104 

8.9 

55841 

20.0 

3200000 

35.5 

56382167 

92 

6590815232 

2.90 

205.111 

9.0 

59049 

20.2 

3363232 

36.0 

60466176 

93 

6956883693 

3.00 

243.000 

9.1 

62403 

20.4 

3533059 

36.5 

64783487 

94 

7339040224 

3.10 

286.292 

9.2 

65908 

20.6 

3709677 

37.0 

69343957 

95 

7737809375 

3.20 

335.544 

9.3 

69569 

20.8 

3893289 

37.5 

74157715 

96 

8153726976 

3.30 

391.354 

9.4 

73390 

21.0 

4084101 

38.0 

79235168 

97 

8587340257 

3.40 

454.354 

9.5 

77378 

21.2 

4282322 

38.5 

84587005 

98 

9039207968 

3.50 

525.219 

9.6 

81537 

21.4 

4488  1  66 

39.0 

90224199 

99 

9509900499 

3.60 

604.662 

9.7 

85873 

21.6 

4701850 

39.5 

96158012 

110 


MATHEMATICAL  TABLES. 


tt 

|- 

53 


1    O 

!l 


81 
Bl 

0 
0 


I 


LO  CO  C 

O  —  r 


-''  — 


--- 


OI>«.O^ 

m  !>•  co  c 

CNt^  — C 


o  o  O  •-  «ri  i 

• 


OvO 
^O  —  C 
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ooow^oooooomom.  oooinoomoooomtnoro  c^.o 
-1  —  !>.  —  mo^-OiAOPOOoor^oot>NO'^-vOfNOOio'Tt' 

-<        —  '- 


«—  —  -•  —  —  «N  «N  <N  (N  <S  (S  (S  fS  <N  C 


,.. 

f^t^.t^.0  —  (NpTiOO'^TOOO  —  <N 

oo  —  covo  —  <NPM  —  ir^rt-r>.in 

O  O  O  O  O  —  fS  CO  vO  00  -O  CO  <N  r 

—  fT 


d  OO  CO  —  ' 

U^ICN  — 


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C.S 


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o  —  oooo 


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Is 

1.- 


CIRCUMFERENCES   AND   AREAS   OF   CIRCLES. 
CIRCUMFERENCES  AND  AREAS  OF  CIRCLES. 


Ill 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

1/64 

.04909 

.00019 

23/8 

7.4613 

4.4301 

6Vs 

19.242 

29  465 

1/32 

.  098  1  8 

.00077 

7/16 

7.6576 

4  .  6664 

1/4 

19.635 

30  680 

3/64 

.14726 

.00173 

i/3 

7.8540 

4.9087 

3/8 

20.028 

31   919 

Vl6 

.19635 

.00307 

9/16 

8.0503 

5.1572 

V2 

20.420 

33    183 

3/32 

.29452 

.00690 

5/8 

8.2467 

5.4119 

5/8 

20.813 

34.472 

Vs 

.39270 

.01227 

H/16 

8.4430 

5.6727 

3/4 

21.206 

35  785 

5/32 

.49087 

.01917 

3/4 

8.6394 

5.9396 

7/8 

21.598 

37.  122 

3/16 

.  58905 

.02761 

13/16 

8.8357 

6.2126 

21.991 

38.485 

7/32 

.  68722 

.03758 

7/8 

9.0321 

6.4918 

*l/8 

22.384 

39.871 

15/16 

9.2284 

6.7771 

V4 

22.776 

41.282 

1/4 

.78540 

.  04909 

3/8 

23  .  1  69 

42.718 

»/32 

.88357 

.06213 

3. 

9.4248 

7.0686 

V2 

23.562 

44.  179 

5/16 

.98175 

.07670 

Vl6 

9.6211 

7.3662 

5/8 

23.955 

45  .  664 

H/32 

.0799 

.09281 

Vs 

9.8175 

7  .  6699 

3/4 

24.347 

47.173 

3/8 

.  1781 

.11045 

3/16 

10.014 

7.9798 

7/8 

24.740 

48.707 

13/32 

.2763 

.12962 

!/4 

10.210 

8.2958 

8. 

25.133 

50.265 

7/16 

.3744 

..15033 

5/16 

10.407 

8.6179 

V8 

25.525 

51.849 

15/32 

.4726 

.17257 

3/8 

10.603 

8.9462 

1/4 

25.918 

53.455 

7/16 

10.799 

9.2806 

3/8 

26.311 

55.088 

1/2 

.5708 

.19635 

1/2 

10.996 

9.6211 

V2 

26.704 

56.745 

17/32 

.6690 

.22166 

9/16 

11.  192 

9.9678 

5/8 

27.096 

58.426 

9/18 

.7671 

.24850 

5/8 

1  1  .  388 

10.321 

3/4 

27.489 

60.  132 

19/32 

.8653 

.27688 

H/16 

11.585 

10.680 

7/8 

27.882 

61.862 

5/8 

.9635 

.30680 

3/4 

11.781 

11.045 

9. 

28.274 

63.617 

21/32 

2.0617 

.33824 

13/16 

11.977 

11.416 

1/8 

28.667 

65.397 

H/16 

2.1598 

.37122 

7/8 

12.174 

1  1  .  793 

1/4 

29.060 

67.201 

23/32 

2.2580 

.40574 

15/16 

12.370 

12.177 

3/8 

29.452 

69.029 

4. 

12.566 

12.566 

1/2 

29.845 

70.882 

3/4 

2.3562 

.44179 

Vl6 

12.763 

12.962 

5/8 

30.238 

72.760 

25/32 

2.4544 

.47937 

1/8 

12.959 

13.364 

3/4 

30.631 

74.662 

13/16 

2.5525 

.51849 

3/16 

13.  155 

13.772 

•7/8 

3  1  .  023 

76.589 

27/32 

2.6507 

.55914 

1/4 

13.352 

14.  186 

10. 

31.416 

78.540 

7/8 

2.7489 

.60132 

5/16 

13.548 

14.607 

V8 

3  1  .  809 

80.516 

29/32 

2.8471 

.64504 

3/8 

13.744 

15.033 

1/4 

32.201 

82.516 

15/16 

2.9452 

.  69029 

7/16 

13.941 

15.466 

3/8 

32.594 

84.541 

31/32 

3.0434 

.73708 

1/2 

14.137 

1  5  .  904 

V2 

32.987 

86.590 

9/16 

14.334 

16.349 

5/8 

33.379 

88,664 

1. 

3.1416 

.7854 

5/8 

14.530 

16.800 

3/4 

33.772 

90.763 

Vl6 

3.3379 

.8866 

U/16 

14.726 

17.257 

7/8 

34.  165 

92.886 

•1/8 

3.5343 

.9940 

3/4 

14.923 

17.721 

11. 

34.558 

95.033 

3/16 

3.7306 

.1075 

13/16 

15.119 

18.  190 

1/8 

34.950 

97.205 

1/4 

3.9270 

.2272 

7/8 

15.315 

18.665 

1/4 

35.343 

99.402 

5/16 

4.1233 

.3530 

15/16 

15.512 

19.147 

3/8 

35.736 

101.62 

3/8 

4.3197 

.4849 

5. 

15.708 

19.635 

V2 

36.  128 

103.87 

7/16 

4.5160 

.6230 

Vl6 

1  5  .  904 

20.  129 

5/8 

36.521 

106.14 

1/2 

4.7124 

.7671 

1/8 

16.  101 

20.629 

3/4 

36.914 

108.43 

'•Vie 

4.9087 

.9175 

3/16 

16.297 

21.135 

7/8 

37.306 

110.75 

5/8 

5.1051 

2.0739 

1/4 

16.493 

21.648 

13. 

37.699 

113.10 

H/16 

5.3014 

2.2365 

5/l6 

16.690 

22  .  1  66 

V8 

38.092 

115.47 

3/4 

5.4978 

2.4053 

3/8 

16.886 

22.691 

1/4 

38.485 

117.86 

13/16 

5.6941 

2.5802 

7/16 

17.082 

23.221 

3/8 

38.877 

120.28 

7/8 

5.8905 

2.7612 

1/2 

17.279 

23.758 

1/2 

39.270 

122.72 

15/16 

6.0868 

2.9483 

9/16 

17.475 

24.301 

5/8 

39.663 

125.19 

5/8 

17.671 

24.850 

3/4 

40.055 

127.68 

3. 

6.2832 

3.1416 

n/ie 

17.868 

25.406 

7/8 

40.448 

130.19 

Vl6 

6.4795 

3.3410 

3/4 

18.064 

25.967 

13. 

40.841 

132.73 

1/8 

6.6759 

3.5466 

13/16 

18.261 

26.535 

V8 

41.233 

135  30 

3/16 

6.8722 

3.7583 

7/8 

18.457 

27.  109 

1/4 

41.626 

137.89 

1/4 

7  .  0686 

3.9761 

15/16 

18.653 

27.688 

3/8 

42.019 

140.50 

5/16 

7.2649 

4.2000 

6. 

18.850 

28.274 

1/2 

42.412 

143.14 

112 


MATHEMATICAL  TABLES. 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

Diam 

Circum. 

Area. 

135/8 

42.804 

145.80 

217/8 

68.722 

375.83 

30  Vs 

94.640 

712.76 

3/4 

43.197 

148.49 

23. 

69.  115 

380.13 

1/4 

95.033 

718.69 

7/8 

43.590 

151.20 

1/8 

69.508 

384.46 

3/8 

95.426 

724  64 

14. 

43  ,  982 

153.94 

1/4 

69.900 

388.82 

1/2 

95.819 

730.62 

1/8 

44.375 

156.70 

3/8 

70.293 

393.20 

5/8 

96.211 

736.62 

1/4 

44.768 

159.48 

1/2 

70.686 

397.61 

3/4 

96.604 

742.64 

3/8 

45.160 

162.30 

5/8 

71.079 

402.04 

7/8 

96.997 

748.69 

V2 

45.553 

165.13 

3/4 

71.471 

406.49 

31. 

97.389 

754.77 

5/8 

45.946 

167.99 

7/8 

71.864 

410.97 

V8 

97.782 

760.87 

3/4 

46.338 

170.87 

23. 

72.257 

415.48 

1/4 

98.175 

766.99 

7/8 

46.731 

173.78 

1/8 

72.649 

420.00 

3/8 

98.567 

773.  14 

15. 

47.124 

176.7! 

1/4 

73.042 

424.56 

1/2 

98.960 

779.31 

J-/8 

47.517 

179.67 

3/8 

73.435 

429.13 

5/8 

99.353 

785.51 

1/4 

47.909 

182.65 

1/2 

73.827 

433.74 

3/4 

99.746 

791    73 

3/8 

48.302 

185.66 

5/8 

74.220 

438.36 

7/8 

100.  138 

797.98 

V2 

48.695 

188.69 

3/4 

74.613 

443.01 

32. 

100.531 

804.25 

5/8 

49.087 

191.75 

7/8 

75.006 

447.69 

Vs 

100.924 

810.54 

3/4 

49.480 

194.83 

24. 

75.398 

452.39 

1/4 

101.316 

816.86 

7/8 

49.873 

197.93 

1/8 

75.791 

457.11 

3/8 

101.709 

823.21 

16. 

50.265 

201.06 

1/4 

76.  184 

461.86 

1/2 

102.102 

829.58 

Vs 

50.658 

204.22 

3/8 

76.576 

466.64 

5/8 

102.494 

835.97 

V4 

51.051 

207.39 

1/2 

76.969 

471.44 

3/4 

102.887 

842.39 

3/8 

51.444 

210.60 

5/8 

77.362 

476.26 

7/8 

103.280 

848.83 

V2 

51.836 

213.82 

3/4 

77.754 

481.11 

33. 

103.673 

855.30 

5/8 

52.229 

217.08 

7/8 

78.147 

485.98 

1/8 

104.065 

861.79 

3/4 

52.622 

220.35 

25. 

78.540 

490.87 

1/4 

104.458 

868.31 

7/8 

53.014 

223.65 

1/8 

78.933 

495.79 

3/8 

104.851 

874.85 

17. 

53.407 

226.98 

1/4 

79.325 

500.74 

V2 

105.243 

881.41 

Vs 

53.800 

230.33 

3/8 

79.718 

505.71 

5/8 

105.636 

888.00 

1/4 

54.192 

233.71 

1/2 

80.111 

510.71 

3/4 

106.029 

894.62 

3/8 

54.585  - 

237.10 

5/8 

80.503 

515.72 

7/8 

106.421 

901.26 

1/2 

54.978 

240.53 

3/4 

80.896 

520.77 

34. 

106.814 

907.92 

5/8 

55.371 

243.98 

7/8 

81.289 

525.84 

V8 

107.207 

914.61 

3/4 

55.763 

247.45 

26. 

81.681 

530.93 

1/4 

107.600 

921.32 

7/8 

56.156 

250.95 

1/8 

82.074 

536.05 

3/8 

107.992 

928.06 

18. 

56.549 

254.47 

1/4 

82.467 

541.19 

1/2 

108.385 

934.82 

1/8 

56.941 

258.02 

3/8 

82.860 

546.35 

5/8 

108.778 

941.61 

1/4 

57.334 

261.59 

1/2 

83.252 

551.55 

3/4 

109.  170 

948.42 

3/8 

57.727 

265.18 

5/8 

83.645 

556.76 

7/8 

109.563 

955.25 

1/2 

58.119 

268.80 

3/4 

-84.038 

562.00 

35. 

109.956 

962  .  1  1 

5/8 

58.512 

272.45 

7/8 

84.430 

567.27 

1/8 

110.348 

969  .  00 

3/4 

58.905 

276.12 

27. 

84.823 

572.56 

1/4 

110.741 

975.91 

7/8 

59.298 

279.81 

Vs 

85.216 

577.87 

3/8 

111.  134 

982.84 

19. 

59.690 

283.53 

1/4 

85.608 

583.21 

1/2 

111.527 

989.80 

1/8 

60.083 

287.27 

3/8 

86.001 

588.57 

5/8 

111.919 

996.78 

1/4 

60.476 

291.04 

1/2 

86.394 

593.96 

3/4 

112.312 

1003.8 

3/8 

60.868 

294.83 

5/8 

86.786 

599.37 

7/8 

112.705 

1010.8 

1/2 

61.261 

298.65 

3/4 

87.179 

604.81 

36. 

113.097 

1017.9 

5/8 

61.654 

302.49 

7/8 

87.572 

610.27 

1/8 

113.490 

1025.0 

3/4 

62  .  046 

306.35 

28. 

87.965 

615.75 

1/4 

113.883 

1032.1 

7/8 

62.439 

310.24 

V8 

88.357 

621.26 

3/8 

114.275 

1039.2 

20. 

62.832 

314.16 

1/4 

88.750 

626.80 

1/2 

114.  668 

1046.3 

1/8 

63.225 

318.10 

3/8 

89.143 

632.36 

5/8 

115.061 

1053.5 

1/4 

63.617 

322.06 

1/2 

89.535 

637.94 

3/4 

115.454 

1060.7 

3/8 

64.010 

326.05 

5/8 

89.928 

643.55 

7/8 

115.846 

1068.0 

1/2 

64.403 

330.06 

3/4 

90.321 

649.18 

37. 

116.239 

1075.2 

5/8 

64.795 

334.10 

7/8 

90.713 

654.84 

1/8 

116.632 

1082.5 

3/4 

65.188 

338.16 

29. 

91.106 

660.52 

1/4 

117.024 

1089.8 

7/8 

65.581 

342.25 

1/8 

91.499 

666.23 

3/8 

117.417 

1097.1 

21. 

65.973 

346.36 

V4 

91  .892 

671.96 

1/2 

17.810 

1104.5 

1/8 

66.366 

350.50 

3/8 

92.284 

677.71 

5/8 

18.202 

1111.8 

1/4 

66.759 

354.66 

1/2 

92.677 

683  .  49 

3/4 

1-8.596 

1119.2 

3/8 

67.152 

358.84 

5/8 

93.070 

689.30 

7/8 

18.988 

1126.7 

1/2 

67.544 

363.05 

3/4 

93.462 

695.13 

38. 

19.381 

1134.1 

5/8 

67.937 

367.28 

7/8 

93.855 

700.98 

1/8 

19.773 

1141.6 

3/4 

68.330 

371.54 

30. 

94.248 

706  .  86 

V4 

120.166 

1149.1 

CIRCUMFERENCES   AND  AREAS   OF   CIRCLES.       113 


Dtam 

Circuin. 

Area. 

Diara 

Circum. 

Area. 

Diam 

Circum. 

Area. 

883/8 

120.559 

1136.6 

465/8 

146.477 

1707.4 

547/g 

172.395 

2365.0 

1/2 

120.951 

1164.2 

3/4 

146.869 

1716.5 

55. 

172.788 

2375.8 

5/8 

121.344 

1171.7 

7/8 

147.262 

1725.7 

1/8 

173.180 

2386.6 

3/4 

121.737 

1179.3 

47. 

147.655 

1734.9 

V4 

173.573 

2397.5 

7/8 

122.129 

1186.9 

Vs 

148.048 

1744.2 

3/8 

173.966 

2408.3 

39. 

122.522 

1194.6 

V4 

148.440 

1753.5 

V2 

174.358 

2419.2 

l/8 

122.915 

1202.3 

3/8 

148.833 

1762.7 

5/8 

174.751 

2430.  1 

1/4 

123.308 

1210.0 

V2 

149.226 

1772.  1 

3/4 

175.  144 

2441.  1 

3/8 

123.700 

1217.7 

5/8 

149.618 

1781.4 

7/8 

175.536 

2452.0 

1/2 

124.093 

1225.4 

3/4 

150.011 

1790.8 

56. 

175.929 

2463.0 

5/8 

124.486 

1233.2 

7/8 

150.404 

1800.  1 

Vs 

176.322 

2474.0 

3/4 

124.878 

1241.0 

48. 

150.796 

1809.6 

1/4 

176.715 

2485.0 

7/8 

125.271 

1248.8 

Vs 

151.189 

1819.0 

3/8 

177.  107 

2496.  1 

40. 

125.664 

1256.6 

1/4 

151.582 

1828.5 

1/2 

177.500 

2507.2 

Vs 

126.056 

1264.5 

$ 

151.975 

1837.9 

5/8 

177.893 

2518.3 

1/4 

126.449 

1272.4 

1/2 

152.367 

1847.5 

3/4 

178.285 

2529.4 

3/8 

126.842 

1280.3 

5/8 

152.760 

1857.0 

7/8 

178.678 

2540.6 

1/9 

127.235 

1288.2 

3/4 

153.  153 

1866.5 

57. 

179.071 

2551.8 

5/8 

127.627 

1296.2 

7/8 

153.545 

1876.  1 

1/8 

1  79.  463 

2563.0 

3/4 

128.020 

1304.2 

49. 

153.938 

1885.7 

V4 

179.856 

2574.2 

7/a 

128.413 

1312.2 

Vs 

154.331 

1895.4 

3/8 

180.249 

2585.4 

41. 

128.805 

1320.3 

V4 

154.723 

1905.0 

1/2 

180.642 

2596.7 

Vs 

129.198 

1328.3 

3/8 

155.116 

1914.7 

5/8 

181.034 

2608.0 

V4 

129.591 

1336.4 

V2 

155.509 

1924.4 

3/4 

181.427 

2619.4 

3/8 

129.983 

1344.5 

5/8 

155.902 

1934.2 

7/8 

181.820 

2630.7 

I/O 

130.376 

1352.7 

3/4 

156.294 

1943.9 

58. 

182.212 

2642.  1 

5/8 

130.769 

1360.8 

7/8 

156.687 

1953.7 

VS 

182.605 

2653.5 

3/4 

131.161 

1369.0 

50. 

157  080 

1963.5 

V4 

182.998 

2664.9 

7/8 

131.554 

1377.2 

Vs 

157.472 

1973.3 

3/8 

183.390 

2676.4 

42. 

131.947 

1385.4 

1/4 

157.865 

1983.2 

1/2 

183.783 

2687.8 

1/8 

132.340 

1393.7 

3/jj 

158.258 

1993.1 

5/8 

184.176 

2699.3 

V? 

132.732 

1  402  .  0 

1/9 

158.650 

2003.0 

3/4 

184.569 

2710.9 

3/8 

133.125 

1410.3 

5/8 

159.043 

2012.9 

7/8 

184.961 

2722.4 

1/9 

133.518 

1418.6 

8/4 

159.436 

2022.8 

59. 

185.354 

2734.0 

5/8 

133.910 

1427.0 

7/8 

159.829 

2032.8 

i/a 

185.747 

2745.6 

3/4 

134.303 

1435.4 

51. 

160.221 

2042.8 

1/4 

186.139 

2757.2 

7/8 

134.696 

1443.8 

Vs 

160.614 

2052.8 

3/8 

186.532 

2768.8 

43. 

135.088 

1452.2 

*/4 

161.007 

2062.9 

1/2 

186.925 

2780.5 

Vs 

135.481 

1460.7 

3/8 

161.399 

2073.0 

5/8 

187.317 

2792.2 

1/4 

135.874 

1469.  1 

1/2 

161.792 

2083.1 

3/4 

187.710 

2803.9 

3/8 

136.267 

1477.6 

5/8 

162.185 

2093.2 

7/8 

188.103 

2815.7 

V2 

136.659 

1486.2 

3/4 

162.577 

2103.3 

60. 

188.496 

2827.4 

5/8 

137.052 

1494.7 

7/8 

162.970 

2113.5 

Vs 

188.888 

2839.2 

3/4 

137.445 

1503.3 

53. 

163.363 

2123.7 

1/4 

189.281 

2851.0 

7/8 

137.837 

1511.9 

Vs 

163.756 

2133.9 

3/8 

189.674 

2862.9 

44. 

138.230 

1520.5 

V4 

164.148 

2144.2 

1/2 

190.066 

2874.8 

Vs 

138.623 

1529.2 

3/8 

164.541 

2154.5 

5/8 

190.459 

2886.6 

V4 

139.015 

1537.9 

1/2 

164.934 

2164.8 

3/4 

190.852 

2898.6 

3/8 

139.408 

1546.6 

5/8 

165.326 

2175.1 

7/8 

191.244 

2910.5 

V2 

139.801 

1555.3 

3/4 

165.719 

2185.4 

61. 

191.637 

2922.5 

5/8 

140.194 

1564.0 

7/8 

166.112 

2195.8 

1/8 

192.030 

2934.5 

3/4 

140.586 

1572.8 

53. 

166.504 

2206.2 

V4 

192.423 

2946.5 

7/8 

140.979 

1581.6 

Vs 

166.897 

2216.6 

3/8 

192.815 

2958.5 

45. 

141.372 

1590.4 

V4 

167.290 

2227.0 

1/2 

193.208 

2970.6 

Vs 

141.764 

1599.3 

3/8 

167.683 

2237.5 

5/8 

193.601 

2982  .  7 

V4 

142.157 

1608.2 

1/2 

168.075 

2248.0 

3/4 

193.993 

2994.8 

3/8 

142.550 

1617.0 

5/8 

1  68  .  468 

2258.5 

7/8 

194.386 

3006.9 

Va 

142.942 

1626.0 

3/4 

168.861 

2269.1 

62. 

194.779^ 

3019.1 

5/8 

143.335 

1634.9 

7/8 

169.253 

2279.6 

Vs 

195.  171 

3031.3 

3/4 

143.728 

1643.9 

54. 

169.646 

2290.2 

V4 

195.564 

3043.5 

7/8 

144.  121 

1652.9 

1/8 

170.039 

2300.8 

3/8 

195.957 

3055.7 

46. 

144.513 

1661.9 

1/4 

170.431 

2311.5 

V2 

196.350 

3068.0 

Vs 

144.906 

1670.9 

3/8 

170.824 

2322.1 

5/8 

196.742 

3080.3 

V4 

145.299 

1680.0 

1/2 

171.217 

2332.8 

8/4 

197.135 

3092.6 

3/8 

145.691 

1689.1 

5/8 

171.609 

2343.5 

7/8 

197.528 

3  1  04  .  9 

,      V2 

146.084 

1698.2 

3/4 

172.002 

2354.3 

63. 

197.920 

3117.2 

114 


MATHEMATICAL  TABLES. 


Diam. 

63V8~ 

Circum. 

Area. 

Diam 

Circum. 

Area. 

Diam 

Circum. 

Area. 
"4979.1 

198.313 

3129.6 

713/8 

224.231 

4001.1 

795/8 

250.149 

1/4 

198.706 

3142.0 

1/2 

224.624 

4015.2 

3/* 

250.542 

4995  .  2 

3/8 

199.098 

3154.5 

5/8 

225.017 

4029.2 

7/8 

250.935 

5010.9 

1/2 

199.491 

3166.9 

3/4 

225.409 

4043.3 

80. 

251.327 

5026.5 

5/8 

199.884 

3179.4 

7/8 

225.802 

4057.4 

1/8 

251.720 

5042.3 

3/4 

200.277 

3191.9 

73. 

226.  195 

4071.5 

1/4 

252.  113 

5058.0 

7/8 

200.669 

3204.4 

Vs 

226.587 

4085.7 

3/8 

252.506 

5073.8 

64. 

201.062 

3217.0 

1/4 

226.980 

4099.8 

!/2 

252.898 

5089.6 

1/8 

201.455 

3229.6 

3/8 

227.373 

4114.0 

5/8 

253.291 

5105.4 

V4 

201.847 

3242.2 

V2 

227.765 

4128.2 

3/4 

253.684 

5121.2 

3/8 

202.240 

3254.8 

5/8 

228.158 

4142.5 

7/8 

254.076 

5137.  1 

Va 

202.633 

3267.5 

3/4 

228.551 

4156.8 

81. 

254.469 

5153.0 

5/8 

203.025 

3280.1 

7/8 

228.944 

4171.1 

V8 

254.862 

5  1  68  .  9 

3/4 

203.418 

3292.8 

73. 

229.336 

4185.4 

1/4 

255.254 

5184.9 

7/8 

203.811 

3305.6 

1/8 

229.729 

4199.7 

3/8 

255.647 

5200.8 

65. 

204.204 

3318.3 

1/4 

230.122 

4214.1 

1/2 

256.040 

5216.8 

Vs 

204.596 

3331.1 

3/8 

230.514 

4228.5 

5/8 

256.433 

5232.8 

V4 

204.989 

3343.9 

V2 

230.907 

4242.9 

3/4 

256.825 

5248.9 

3/8 

205.382 

3356.7 

5/8 

231.300 

4257.4 

7/8 

257.218 

5264.9 

V2 

205.774 

3369.6 

3/4 

231.692 

4271.8 

83. 

257.611 

5281.0 

5/8 

206.167 

3382.4 

7/8 

232.085 

4286.3 

1/8 

258.003 

5297.  1 

3/4 

206.560 

3395.3 

74. 

232.478 

4300.8 

1/4 

258.396 

5313.3 

7/8 

206.952 

3408.2 

1/8 

232.871 

4315.4 

3/8 

258.789 

5329.4 

66. 

207.345 

3421.2 

1/4 

233.263 

4329.9 

1/2 

259.181 

5345.6 

1/8 

207.738 

3434.2 

3/8 

233.656 

4344.5 

5/8 

259.574 

5361.8 

1/4 

208.131 

3447.2 

1/2 

234.049 

4359.2 

3/4 

259.967 

5378.  1 

3/8 

208.523 

3460.2 

5/8 

234.441 

4373.8 

7/8 

260.359 

5394.3 

1/2 

208.916 

3473.2 

3/4 

234.834 

4388.5 

83. 

260.752 

5410.6 

5/8 

209.309 

3486.3 

7/8 

235.227 

4403  .  1 

1/8 

261.145 

5426.9 

3/4 

209.701 

3499.4 

75. 

235.619 

4417.9 

l/4 

261.538 

5443.3 

7/8 

210.094 

3512.5 

1/8 

236.012 

4432.6 

3/8 

261.930 

5459.6 

67. 

210.487 

3525.7 

1/4 

236.405 

4447.4 

1/2 

262.323 

5476.0 

1/8 

210.879 

3538.8 

3/8 

236.798 

4462.2 

5/8 

262.716 

5492.4 

1/4 

211.272 

3552.0 

V2 

237.190 

4477.0 

3/4 

263.108 

5508.8 

3/8 

211.665 

3565.2 

5/8 

237.583 

-^491.8 

7/8 

263.501 

5525.3 

1/2 

212.058 

3578.5 

3/4 

237.976 

4506.7 

84. 

263.894 

5541.8 

5/8 

212.450 

3591.7 

7/8 

238.368 

4521.5 

1/8 

264.286 

5558.3 

3/4 

212.843 

3605.0 

76c 

238.761 

4536.5 

V4 

264.679 

5574.8 

7/8 

213.236 

3618.3 

1/8 

239.154 

4551.4 

3/8 

265.072 

5591.4 

68. 

213.628 

3631.7 

V4 

239.546 

4566.4 

1/2 

265.465 

5607.9 

Vs 

214.021 

3645.0 

3/8 

239.939 

4581.3 

5/8 

265.857 

5624.5 

V4 

214.414 

3658.4 

1/2 

240.332 

4596.3 

3/4 

266.250 

5641.2 

3/8 

214.806 

3671.8 

5/8 

240.725 

4611.4 

7/8 

266.643 

5657.8 

1/2 

215.199 

3685.3 

3/4 

241.  117 

4626.4 

85. 

267.035 

5674.5 

5/8 

215.592 

3698.7 

7/8 

241.510 

4641.5 

1/8 

267.428 

5691.2 

3/4 

2  1  5  .  984 

3712.2 

77. 

241.903 

4656.6 

V4 

267.821 

5707.9 

7/8 

216.377 

3725.7 

1/8 

242.295 

4671.8 

3/8 

268.213 

'5724.7 

69. 

216.770 

3739.3 

1/4 

242.688 

4686.9 

V2 

268.606 

5741.5 

Vs 

217.163 

3752.8 

3/8 

243.081 

4702.1 

5/8 

268.999 

5758.3 

V4 

217.555 

3766.4 

1/2 

243.473 

4717.3 

3/4 

269.392 

5775.1 

3/8 

2  1  7  .  948 

3780.0 

5/8 

243.866 

4732.5 

7/8 

269.784 

5791.9 

1/2 

218.341 

3793.7 

3/4 

244.259 

4747.8 

86. 

270.177 

5808.8 

5/8 

218.733 

3807.3 

7/8 

244.652 

4763.1 

1/8 

270.570 

5825.7 

3/4 

219.126 

3821.0 

78. 

245.044 

4778.4 

1/4 

270.962 

5842.6 

7/8 

219.519 

3834.7 

1/8 

245.437 

4793.7 

3/8 

271.355 

5859.6 

70. 

219.911 

3848.5 

1/4 

245.830 

4809.0 

1/9 

271.748 

5876.5 

V8 

220.304 

3862.2 

3/8 

246.222 

4824.4 

5/J 

272.140 

5893.5 

1/4 

220.697 

3876.0 

i/2 

246.615 

4839.8 

3/4 

272.533 

5910.6 

3/8 

221.090 

3889.8 

5/8 

247.008 

4855.2 

7/8 

272.926 

5927.6 

V2 

221.482 

3903.6 

3/4 

247   400 

4870.7 

87. 

273.319 

5944.7 

5/8 

221.375 

3917.5 

7/8 

247.793 

4886.2 

Vs 

273.711 

5961.8 

3/4 

222.268 

3931.4 

79. 

248.  186 

4901.7 

1/4 

274.  104 

5978.9 

7/8 

222  .  660 

3945.3 

!/8 

248.579 

4917.2 

3/8 

274.497 

5996.0 

71. 

223.053 

3959  2 

1/4 

248.971 

4932.7 

1/2 

274.889 

6013.2 

V8 

223  .  446 

3973   1 

3/8 

249.364 

4948.3 

5/8 

275.282 

6030.4 

1/4 

223.838 

3987.1 

V2 

249.757 

4963.9 

3/4 

275.675 

6047.6 

CIRCUMFERENCES    AND   AREAS   OF   CIRCLES.       115 


Diani 

Circum. 

Area. 

Diam 

Circum. 

Area. 

Diam 

Circum. 

Area. 

877/s 

276.067 

6064.9 

957/s 

301.200 

7219.4 

130 

408.41 

13273.23 

88. 

276.460 

6082.1 

96. 

301.593 

7238.2 

131 

411.55 

13478.22 

1/8 

276.853 

6099.4 

1/8 

301.986 

7257.1 

132 

414.69 

13684.78 

1/4 

277.246 

6116.7 

1/4 

302.378 

7276.0 

133 

417.83 

13892.91 

3/8 

277.638 

6134.1 

3/8 

302.771 

7294.9 

134 

420.97 

1  4  1  02  .  6  1 

1/2 

278.031 

6151.4 

1/2 

303.  164 

7313.8 

135 

424.  12 

14313.88 

5/8 

278.424 

6168.8 

5/8 

303.556 

7332.8 

136 

427.26 

14526.72 

3/4 

278.816 

6186.2 

3/4 

303.949 

7351.8 

137 

430.40 

!4741  .  14 

7/8 

279.209 

6203  .  7 

7/8 

304.342 

7370.8 

138 

433.54 

14957.12 

89. 

279.602 

6221.1 

97. 

304.734 

7389.8 

139 

436.68 

15174.68 

V8 

279.994 

6238.6 

1/8 

305.  127 

7408.9 

140 

439.82 

15393.80 

1/4 

280.387 

6256.1 

1/4 

305.520 

7428.0 

141 

442.96 

15614.50 

3/8 

280.780 

6273.7 

3/8 

305.913 

7447.1 

142 

446.  11 

15836.77 

Vz 

281.  173 

6291.2 

1/2 

306.305 

7466.2 

143 

449.25 

16060.61 

5/8 

281.565 

6308.8 

5/8 

306.698 

7485.3 

144 

452.39 

16286.02 

3/4 

281.958 

6326.4 

3/4 

307.091 

7504.5 

145 

455.53 

16513.00 

7/8 

282.351 

6344.1 

7/8 

307.483 

7523.7 

146 

458.67 

16741.55 

90. 

282.743 

6361.7 

98. 

307.876 

7543.0 

147 

461.81 

16971.67 

Vs 

283.136 

6379.4 

1/8 

308.269 

7562.2 

148 

464.96 

17203.36 

1/4 

283.529 

6397.1 

1/4 

308.661 

7581.5 

149 

468.10 

17436.62 

3/8 

283.921 

64  1  4  .  9 

3/8 

309.054 

7600.8 

150 

471.24 

17671.46 

1/2 

284.314 

6432.6 

1/2 

309.447 

7620.  1 

151 

474.38 

17907.86 

5/8 

284.707 

6450.4 

5/8 

309.840 

7639.5 

152 

477.52 

18145.84 

3/4 

285.100 

6468.2 

3/4 

310.232 

7658.9 

153 

480.66 

18385.39 

7/8 

285.492 

6486.0 

7/8 

310.625 

7678.3 

154 

483.81 

18626.50 

91. 

285.885 

6503.9 

99. 

311.018 

7697.7 

155 

486.95 

18869.19 

Vs 

286.278 

6521.8 

1/8 

311.410 

7717.1 

156 

490  .  09 

19113.45 

1/4 

286.670 

6539.7 

1/4 

311.803 

7736.6 

157 

493.23 

19359.28 

3/8 

287.063 

6557.6 

3/8 

312.  196 

7756.1 

158 

496.37 

19606.68 

V2 

287.456 

6575.5 

1/2 

312.588 

7775.6 

159 

499.51 

19855.65 

5/8 

287.848 

6593.5 

5/8 

312.981 

7795.2 

160 

502.65 

20106.  19 

3/4 

288.241 

6611.5 

3/4 

313.374 

7814.8 

161 

505  .  80 

20358.31 

7/8 

288.634 

6629.6 

7/8 

313.767 

7834.4 

162 

508.94 

20611.99 

93. 

289.027 

6647.6 

100 

314".  159 

7854.0 

163 

512.08 

20867.24 

Vs 

289.419 

6665  .  7 

101 

317.30 

8011  .85 

164 

515.22 

21124.07 

V4 

289.812 

6683.8 

102 

320.44 

8171.28 

165 

518.36 

21382.46 

3/8 

290.205 

6701.9 

103 

323.58 

8332.29 

166 

521.50 

21642.43 

V2 

290.597 

6720.  1 

104 

326.73 

8494.87 

167 

524.65 

21903.97 

5/8 

290.990 

6738.2 

105 

329.87 

8659.01 

168 

527.79 

22167.08 

3/4 

291.383 

6756.4 

106 

333.01 

8824.73 

169 

530.93 

22431.76 

7/8 

291.775 

6774.7 

107 

336.15 

8992.02 

170 

534.07 

22698.01 

93. 

292.168 

6792.9 

108 

339.29 

9160.88 

171 

537.21 

22965  .  83 

V8 

292.561 

6811.2 

109 

342.43 

9331.32 

172 

540.35 

23235.22 

V4 

292.954 

6829.5 

110 

345.58 

9503.32 

173 

543.50 

23506.18 

3/8 

293.346 

6847.8 

111 

348.72 

9676.89 

174 

546.64 

23778.71 

1/2 

293.739 

6866.1 

112 

351.86  > 

9852.03 

175 

549.78 

24052.82 

5/8 

294.  132 

6884.5 

113 

355.00 

0028.75 

176 

552.92 

24328.49 

3/4 

294.524 

6902  .  9 

114 

358.14 

0207.03 

177 

556.06 

24605  .  74 

7/8 

294.917 

6921.3 

115 

361.28 

0386.89 

178 

559.20 

24884.56 

94. 

295.310 

6939.8 

116 

364.42 

0568.32 

179 

562.35 

25164.94 

1/8 

295  .  702 

6958.2 

117 

367.57 

0751.32 

ISO 

565  .  49 

25446.90 

1/4 

296.095 

6976.7 

118 

370.71 

0935.88 

181 

568.63 

25730.43 

3/8 

296.488 

6995.3 

119 

373.85 

1122.02 

182 

571.77 

26015.53 

1/2 

296.881 

701.3.8 

120 

376.99 

1309.73 

183 

574.91 

26302.20 

5/8 

297.273 

7032.4 

121 

380.13 

1499.01 

184 

578.05 

26590.44 

3/4 

297.666 

7051.0 

122 

383.27 

1689.87 

185 

581.19 

26880.25 

7/8 

298.059 

7069.6 

123 

386.42 

1882.29 

186 

584.34 

27171.63 

95. 

298.451 

7088.2 

124 

389.56 

2076.28 

187 

587.48 

27464.59 

1/8 

298.844 

7106.9 

125 

392.70 

2271.85 

188 

590.62 

27759.11 

1/4 

299.237 

7125.6 

126 

395.84 

2468.98 

189 

593  .  76 

28055.21 

3/8 

299.629 

7144.3 

127 

398.98 

2667.69 

190 

596.90 

28352.87 

1/2 

300.022 

7163.0 

128 

402.  12 

2867.96 

191 

600.04 

28652.  11 

5/8 

300.415 

7181.8 

129 

405.27 

3069.81 

192 

603.19 

28952.92 

3/4 

•  n      i 

300.807 

7200.6 

116 


MATHEMATICAL  TABLES. 


Diam 

Circum 

Area. 

Diam 

Circum 

Area. 

Diam 

Circum. 

Area. 

193 

606.33 

29255.30 

260 

816.81 

53092.92 

327 

1027.30 

83961.84 

194 

609.47 

29559.25 

261 

819.96 

53502.11 

328 

1030.44 

84496.28 

195 

612.61 

29864.77 

262 

823.10 

53912.87 

329 

1033.58 

85012.28 

196 

615.75 

30171.86 

263 

826.24 

54325.  21 

330 

1036.73 

85529.86 

197 

618.89 

30480.52 

264 

829.38 

54739.11 

331 

1039.87 

86049.01 

193 

622.04 

30790.75 

265 

832.52 

55154.59 

332 

1043.01 

86569.73 

199 

625.  18 

31102.55 

266 

835.66 

55571.63 

333 

1046.  15 

87092.02 

200 

628.32 

31415.93 

267 

838.81 

55990.25 

334 

1049.29 

87615.88 

201 

631.46 

31730.87 

268 

841.95 

56410.44 

335 

1052.43 

88141.31 

202 

634.60 

32047.39 

269 

845.09 

56832.20 

336 

1055.58 

88668.31 

203 

637.74 

32365.47 

270 

848.23 

57255.53 

337 

1058.72 

89196.88 

204 

640.88 

32685.  13 

271 

851.37 

57680.43 

338 

1061  .86 

89727.03 

205 

644.03 

33006.36 

272 

854.51 

58106.90 

339 

1065.00 

90258.74 

206 

647.  17 

33329.  16 

273 

857.65 

58534.94 

340 

1068.  14 

90792.03 

207 

650.31 

33653.53 

274 

860.80 

58964.55 

341 

1071.28 

91326.88 

203 

653.45 

33979.47 

275 

863  .  94 

59395.74 

342 

1074.42 

91863.31 

209 

656.59 

34306.98 

276 

867.08 

59828.49 

343 

1077.57 

92401.31 

210 

659.73 

34636.06 

277 

870.22 

60262.82 

344 

1080.71 

92940.88 

211 

662.88 

34966.71 

278 

873.36 

60698.71 

345 

1083.85 

93482.02 

212 

666.02 

35298.94 

279 

876.50 

61136.  18 

346 

1086.99 

94024.73 

213 

669.  16 

35632.73 

280 

879.65 

61575.22 

347 

1090.  13 

94569.01 

214 

672.30 

35968.09 

281 

882.79 

62015.82 

348 

1093.27 

95114.86 

215 

675.44 

36305.03 

282 

885.93 

62458.00 

349 

1096.42 

95662.28 

216 

678.58 

36643.54 

283 

889.07 

62901.75 

350 

1099.56 

96211.28 

217 

681.73 

36983.61 

284 

892.21 

63347.07 

351 

1102.70 

96761.84 

218 

684.87 

37325.26 

285 

895.35 

63793.97 

352 

1105.84 

97313.97 

219 

688.01 

37668.48 

286 

898.50 

64242.43 

353 

1108.98 

97867.68 

230 

691.  15 

38013.27 

287 

901.64 

64692.46 

354 

1112.  12 

98422.96 

221 

694.29 

38359.63 

288 

904.78 

65144.07 

355 

1115.27 

98979.80 

222 

697.43 

38707.56 

289 

907.92 

65597.24 

356 

1118.41 

99538.22 

223 

700.58 

39057.07 

290 

911.06 

66051.99 

357 

1121.55 

100098.21 

224 

703.72 

39408.  14 

291 

914.20 

66508.30 

358 

1124.69 

100659.77 

225 

706.86 

39760.78 

292 

917.35 

66966.  19 

359 

1127.83 

101222.90 

226 

710.00 

401  15.00 

293 

920.49 

67425.65 

360 

1130.97 

101787.60 

227 

713.  14 

40470.78 

294 

923.63 

67886.68 

361 

1134.11 

102353.87 

228 

716.28 

40828.  14 

295 

926.77 

68349.28 

362 

1137.26 

102921.72 

229 

719.42 

41187.07 

296 

929.91 

68813.45 

363 

1140.40 

103491.  13 

230 

722.57 

41547.56 

297 

933.05 

69279.  19 

364 

1143.54 

1  04062  .  1  2 

231 

725.71 

41909.63 

298 

936.  19 

69746.50 

365 

1146.68 

104634.67 

232 

728.85 

42273.27 

299 

939.34 

70215.38 

366 

1149.82 

105208.80 

233 

73  1  .  99 

42638.48 

300 

942.48 

70685.83 

367 

1152.96 

105784.49 

234 

735.13 

43005.26 

301 

945.62 

71157.86 

368 

1156.11 

106361.76 

235 

738.27 

43373.61 

302 

948.76 

71631.45 

369 

1159.25 

106940.60 

236 

741.42 

43743.54 

303 

951.90 

72106.62 

370 

1162.39 

107521.01 

237 

744.56 

44115.03 

304 

955.04 

72583.36 

371 

1165.53 

108102.99 

238 

747.70 

44488.09 

305 

958.  19 

73061.66 

372 

1  168.67 

108686.54 

239 

750.84 

44862.73 

306 

961.33 

73541  .54 

373 

1171.81 

109271.66 

240 

753.98 

45238.93 

307 

964.47 

74022.99 

374 

1174.96 

109858.35 

241 

757.  12 

45616.71 

308 

967.61 

74506.01 

375 

1178.10 

1  10446.62 

242 

760.27 

45996.06 

309 

970.75 

74990.60 

376 

1181.24 

111036.45 

243 

763.41 

46376.98 

310 

973.89 

75476.76 

377 

1184.33 

1  1  1627.86 

244 

766.55 

46759.47 

311 

977.04 

75964.50 

378 

1187.52 

112220.83 

245 

769.69 

47143.52 

312 

980.  18 

76453.80 

379 

1190.66 

112815.38 

246 

772.  S3 

47529.  16 

313 

983.32 

76944.67 

380 

1193.81 

113411.49 

247 

775.97 

47916.36 

314 

986.46 

77437.12 

381 

1196.95 

114009.18 

248 

779.  11 

48305.13 

315 

989.60 

77931.  13 

382 

1200.09 

1  14608.44 

249 

782.26 

48695.47 

316 

992  .  74 

78426.72 

383 

1203.23 

115209.27 

250 

785.40 

49087.39 

317 

995.88 

78923.88 

384 

1206.37 

115811;67 

251 

788.54 

49480.87 

318 

999.03 

79422.60 

385 

1209.51 

116415.64 

252 

791.68 

49875.92 

319 

1002.17 

79922  .  90 

386 

1212.65 

117021.18 

253 

794.82 

50272.55 

320 

1005.31 

80424.77 

387 

1215.80 

117628.30 

254 

797.96 

50670.75 

321 

1008.45 

80928.21 

388 

1218.94 

18236.98 

255 

801.  11 

51070.52 

322 

1011.59 

81433.22 

389 

1222.08 

18847.24 

256 

804.25 

51471.85 

323 

1014.73 

81939.80 

390 

1225.22 

19459.06 

257 

807.39 

51874.76 

324 

1017.88 

82447.96 

391 

1228.36 

20072.46 

258 

810.53 

52279.24 

325 

1021.02 

82957.68 

392 

1231.50 

20687.46 

259 

813.67 

52685  .  29 

326 

1024.  16 

83468.98 

393 

1234.65 

21303.96 

CIRCUMFERENCES  AND   AREAS  OF  CIRCLES. 


117 


Diam 

Circum 

Area. 

Diam 

Circum 

Area. 

Diam 

Circum 

Area. 

"394" 

1237.79 

121922.07 

461 

1448.27 

166913.60 

528 

1658.76 

218956.44 

395 

1240.93 

122541.75 

462 

1451.42 

167638.53 

529 

1661.90 

219786.61 

396 

1244.07 

123163.00 

463 

1454.56 

168365.02 

530 

1665.04 

220618.34 

397 

1247.21 

123785.82 

464 

1457.70 

1  69093  .  08 

531 

1668.  19 

221451.65 

398 

1250.35 

124410.21 

465 

1460.84 

169822.72 

532 

1671.33 

222286.53 

399 

1253.50 

125036.  17 

466 

1463.98 

170553.92 

533 

1674.47 

223122.98 

400 

1256.64 

125663.71 

467 

1467.  12 

171286.70 

534 

1677.61 

223961.00 

401 

1259.78 

126292.81 

468 

1470.27 

172021.05 

535 

1680.75 

224800.59 

402 

1262.92 

126923.48 

469 

1473.41 

172756.97 

536 

1683.89 

225641.75 

403 

1266.06 

127555.73 

470 

1476.55 

173494.45 

537 

1687.04 

226484.48 

404 

1269.20 

128189.55 

471 

1479.69 

174233.51 

538 

1690.18 

227328.79 

405 

1272.35 

128824.93 

472 

1482.83 

174974.  14 

539 

1693.32 

228174.66 

406 

1275.49 

129461  .89 

473 

1485.97 

175716.35 

540 

1696.46 

229022.  10 

407 

1278.63 

130100.42 

474 

1489.11 

176460.12 

541 

1699.60 

229871.12 

408 

1281.77 

130740.52 

475 

1492.26 

177205.46 

542 

1702.74 

230721.71 

409 

1284.91 

131382.  19 

476 

1495.40 

177952.37 

543 

1705.88 

231573.86 

410 

1288.05 

132025.43 

477 

1498.54 

178700.86 

544 

1709.03 

232427.59 

411 

1291.  19 

132670.24 

478 

1501.68 

179450.91 

545 

1712.17 

233282.89 

412 

1294.34 

133316.63 

479 

1504.82 

180202.54 

546 

1715.31 

234139.76 

413 

1297.48 

133964.58 

480 

1507.96 

180955.74 

547 

1718.45 

234998.20 

414 

1300.62 

134614.10 

481 

1511.11 

181710.50 

548 

1721.59 

235858.21 

415 

1303.76 

135265.20 

482 

1514.25 

182466.84 

549 

1724.73 

236719.79 

416 

1306.90 

135917.86 

483 

1517.39 

183224.75 

550 

1727.88 

237582.94 

417 

1310.04 

136572.10 

484 

1520.53 

183984.23 

551 

1  73  1  .  02 

238447.67 

418 

1313.19 

137227.91 

485 

1523.67 

184745.28 

552 

1734.16 

239S13.96 

419 

1316.33 

137885.29 

486 

1526.81 

185507.90 

553 

1737.30 

240181.83 

420 

1319.47 

138544.24 

487 

1529.96 

186272.10 

554 

1740.44 

241051.26 

421 

1322.61 

139204.76 

488 

1533.10 

187037.86 

555 

1743.58 

241922.27 

421 

1325.75 

139866.85 

489 

1536.24 

187805.19 

556 

1746.73 

242794.85 

423 

1328.89 

140530.51 

49O 

1539.38 

188574.10 

557 

1749.87 

243668.99 

424 

1332.04 

141195.74 

491 

1542.52 

169344.5. 

558 

1753.01 

244544.71 

425 

1335.  18 

141862.54 

492 

1545.66 

1901  16.62 

559 

1756.15 

245422.00 

426 

1338.32 

142530.92 

493 

1548.81 

190890.2 

560 

1759.29 

246300.86 

427 

1341.46 

143200.86 

494 

1551.95 

191665.43 

561 

1  762  .  43 

247181.30 

428 

1344.60 

143872.38 

495 

1555.09 

192442.  18 

562 

1765.58 

248063.30 

429 

1347.74 

1  44545.  46 

496 

1558.23 

193220.51 

563 

1768.72 

248946.87 

430 

1350.88 

145220.  12 

497 

1561.37 

194000.41 

564 

1771  .86 

249832.01 

431 

1354.03 

145896.35 

498 

1564.51 

194781.89 

565 

1775.00 

250718.73 

432 

1357.17 

146574.15 

499 

1567.65 

195564.93 

566 

1778.14 

251607.01 

433 

1360.31 

147253.52 

500 

1570.80 

196349.54 

567 

1781.28 

252496.87 

434 

1363.45 

147934.46 

501 

1573.94 

197135.72 

568 

1784.42 

253388.30 

435 

1366.59 

148616.97 

502 

1577.03 

197923.48 

569 

1787.57 

254281.29 

436 

1369.73 

149301.05 

503 

1580.22 

198712.80 

570 

1790.71 

255175.86 

437 

1372  88 

149986.70 

504 

1583.36 

199503.70 

571 

1793.85 

256072.00 

438 

1376.02 

150673.93 

505 

1586.50 

200296.17 

572 

1796.99 

256969.71 

439 

1379.  16 

151362.72 

506 

1589.65 

201090.20 

573 

1800.  13 

257868.99 

440 

1382.30 

152053.08 

507 

1592.79 

201885.81 

574 

1803.27 

258769.85 

441 

1385.44 

152745.02 

508 

1595.93 

202682.99 

575 

1806.42 

259672.27 

442 

1388.58 

153438.53 

509 

1599.07 

203481.74 

576 

1809.56 

260576.26 

443 

1391.73 

154133.60 

510 

1602.21 

204282.06 

577 

1812.70 

261481.83 

444 

1394.87 

154830.25 

511 

1605.35 

205083.95 

578 

1815.84 

262388.96 

445 

1398.01 

155528.47 

512 

1608.50 

205887.42 

579 

1818.93 

263297.67 

446 

1401.  15 

156228.26 

513 

1611.64 

206692.45 

580 

1822.12 

264207.94 

447 

1404.29 

156929.62 

514 

1614.78 

207499.05 

581 

1825.27 

265119.79 

448 

1407.43 

157632.55 

515 

1617.92 

208307.23 

582 

1828.41 

266033.21 

449 

1410.58 

158337.06 

516 

1  62  1  .  06 

209116.97 

583 

1831.55 

266948.20 

450 

1413.72 

159043.13 

517 

1624.20 

209928.29 

584 

1834.69 

267864.76 

451 

1  4  1  6  .  86 

159750.77 

518 

1627.34 

210741.18 

585 

1837.83 

268782.89 

452 

1420.00 

160459.99 

519 

1630.49 

211555.63 

586 

1840.97 

269702.59 

453 

1423.14 

161170.77 

520 

1633.63 

212371.66 

587 

1844.11 

270623.86 

454 

1426.28 

161883.  13 

521 

1636.77 

213189.26 

588 

1847.26 

271546.70 

455 

1429.42 

162597.05 

522 

1639.91 

214008.43 

589 

1850.40 

272471.  12 

456 

1432.57 

163312.55 

523 

1643.05 

214829.  17 

59O 

1853.54 

273397.10 

457 

1435.71 

164029.62 

524 

1646.  19 

215651.49 

591 

1856.68 

274324.66 

458 

1438  85 

164748  26 

525 

1649.34 

216475.37 

592 

1859.82 

275253.78 

459 
460 

1441.99 
1445.13 

165468.47 
166190.25 

526 
527 

1652.48 
1655.62 

217300.82 
218127.85 

593 
594 

1862.96  276184.48 
1866.1l'277l16.75 

118 


MATHEMATICAL  TABLES. 


Diam 

Circum. 

Area. 

Diam 

Circum, 

Area. 

Diam 

Circum 

Area. 

595 

1869.25 

278050.58 

663 

2082.88 

345236.69 

731 

2296.50 

419686.  13 

596 

1872.39 

278985.99 

664 

2086.02 

346278.91 

732 

2299.65 

420835  19 

597 

1875.53 

279922.97 

665 

2089.16 

347322.70 

733 

2302.79 

421985*  79 

598 

1878.67 

280861.52 

666 

2092.30 

348368.07 

734 

2305.93 

423137.97 

599 

1881.81 

281801.65 

667 

2095.44 

349415.00 

735 

2309.07 

424291.72 

600 

1884.96 

282743.34 

668 

2098.58 

350463.51 

736 

2312.2 

425447.04 

601 

1888.  10 

283686.60 

669 

2101.73 

351513.59 

737 

2315.35 

426603  .  94 

602 

1891.24 

284631.44 

670 

2104.87 

352565.24 

738 

2318.50 

427762.40 

603 

1894.38 

285577.84 

671 

2108.01 

353618.45 

739 

2321.64 

428922.43 

604 

1897.52 

286525.82 

672 

2111.15 

354673.24 

740 

2324.78 

430084.03 

605 

1900.66 

287475.36 

673 

2114.29 

355729.60 

741 

2327.92 

431247.21 

606 

1903.81 

288426.48 

674 

2117.43 

356787.54 

742 

233  1  .  06 

432411.95 

607 

1906.95 

289379.17 

675 

2120.58 

357847.04 

743 

2334.20 

433578.27 

608 

1910.09 

290333.43 

676 

2123.72 

358908.11 

744 

2337.34 

434746.  \6 

609 

1913.23 

291289.26 

677 

2126.86 

359970.75 

745 

2340.49 

435915.62 

610 

1916.37 

292246.66 

678 

2130.00 

361034.97 

746 

2343.63 

437086.64 

611 

1919.51 

293205.63 

679 

2133.14 

362100.75 

747 

2346.77 

438259.24 

612 

1922.65 

294166.17 

680 

2136.28 

363168.11 

748 

2349.91 

439433.41 

613 

1925.80 

295128.28 

681 

2139.42 

364237.04 

749 

2353.05 

440609.16 

614 

1928.94 

296091.97 

682 

2142.57 

365307.54 

750 

2356.19 

441786.47 

615 

1932.08 

297057.22 

683 

2145.71 

366379.60 

751 

2359.34 

442965.35 

616 

1935.22 

298024.05 

684 

2148.85 

367453.24 

752 

2362.48 

444145.80 

617 

1938.36 

298992.44 

685 

2151.99 

368528.45 

753 

2365.62 

445327.83 

618 

1941.50 

299962.41 

686 

2155.13 

369605.23 

754 

2368.76 

446511.42 

619 

1944.65 

300933.95 

687 

2158.27 

370683.59 

755 

2371.90 

447696.59 

620 

1947.79 

301907.05 

688 

2161.42 

371763.51 

756 

2375.04 

448883.32 

621 

1950.93 

302881.73 

689 

2164.56 

372845.00 

757 

2378.19 

450071.63 

622 

1954.07 

303857.98 

690 

2167.70 

373928.07 

758 

2381.33 

451261.51 

623 

1957.21 

304835.80 

691 

2170.84 

375012.70 

759 

2384.47 

452452.96 

624 

1960.35 

305815.20 

692 

2173.98 

376098.91 

760 

2387.61 

453645.98 

625 

1963.50 

306796.16 

693 

2177.12 

377186.68 

761 

2390.75 

454840.57 

626 

1966.64 

307778.69 

694 

2180.27 

378276.03 

762 

2393.89 

456036.73 

627 

1969.78 

308762.79 

695 

2183.41 

379366.95 

763 

2397.04 

457234.46 

628 

1972.92 

309748.47 

696 

2186.55 

380459.44 

764 

2400.18 

458433.77 

629 

1976.06 

310735.71 

697 

2189.69 

381553.50 

765 

2403.32 

459634.64 

630 

1979.20 

311724.53 

698 

2192.83 

382649.13 

766 

2406.46 

460837.08 

631 

1982.35 

312714.92 

699 

2195.97 

383746.33 

767 

2409.60 

462041.10 

632 

1985.49 

313706.88 

700 

2199.11 

384845.10 

768 

2412.74 

463246.69 

633 

1988.63 

314700.40 

701 

2202.26 

385945.44 

769 

2415.88 

464453.84 

634 

1991.77 

315695.50 

702 

2205.40 

387047.36 

770 

2419.03 

465662.57 

635 

1994.91 

316692.17 

703 

2208.54 

388150.84 

771 

2422.17 

466872.87 

636 

1998.05 

317690.42 

704 

2211.68 

389255.90 

772 

2425.31 

468084.74 

637 

2001.  19 

318690.23 

705 

2214.82 

390362.52 

773 

2428.45 

469298.  18 

638 

2004.34 

319691.61 

706 

2217.96 

391470.72 

774 

2431.59 

470513.19 

639 

2007.48 

320694.56 

707 

2221.11 

392580.49 

775 

2434.73 

471729.77 

640 

2010.62 

321699.09 

708 

2224.25 

393691.82 

776 

2437.88 

472947.92 

641 

2013.76 

322705.18 

709 

2227.39 

394804.73 

777 

2441.02 

474167.65 

642 

2016.90 

323712.85 

710 

2230.53 

395919.21 

778 

2444.16 

475388.94 

643 

2020.04 

324722.09 

711 

2233.67 

397035.26 

779 

2447.30 

476611.81 

644 

2023.  19 

325732.89 

712 

2236.81 

398152.89 

780 

2450.44 

477836.24 

645 

2026.33 

326745.27 

713 

2239.96 

399272.08 

781 

2453.58 

479062.25 

646 

2029.47 

327759.22 

714 

2243.10 

400392.84 

782 

2456.73 

480289.83 

647 

2032.61 

328774.74 

715 

2246.24 

401515.18 

783 

2459.87 

481518.97 

648 

2035.75 

329791.83 

716 

2249.38 

402639.08 

784 

2463.01 

482749.69 

649 

2038.89 

330810.49 

717 

2252.52 

403764.56 

785 

2466.15 

483981.98 

650 

2042.04 

331830.72 

718 

2255.66 

404891.60 

786 

2469.29 

485215.84 

651 

2045.  18 

332852.53 

719 

2258.81 

406020.22 

787 

2472.43 

48645  1  .  28 

652 

2048.32 

333875.90 

720 

2261.95 

407150.41 

788 

2475.58 

487688.28 

653 

2051.46 

334900.85 

721 

2265  .  09 

408282.17 

789 

2478.72 

488926.85 

654 

2054.60 

335927.36 

722 

2268.23 

409415.50 

790 

2481.86 

490166.99 

655 

2057.74 

336955.45 

723 

2271.37 

410550.40 

791 

2485.00 

491408.71 

656 

2060.88 

337985.10 

724 

2274.51 

411686.87 

792 

2488.14 

492651.99 

657 

2064.03 

339016.33 

725 

2277.65 

412824.91 

793 

2491.28 

493896.85 

.  658 

2067.17 

340049.13 

726 

2280.80 

413964.52 

794 

2494.42 

495143.28 

659 

2070.31 

341083.50 

727 

2283.94 

415105.71 

795 

2497.57 

496391.27 

660 

2073.45 

342119.44 

728 

2287.085416248.46 

796 

2500.71 

497640.84 

661 

2076.59 

343156.95 

729 

2290.22 

417392.79 

797 

2503.85 

498891.98 

£62 

2079.73  344196.03 

730  2293.36 

418538.68 

798 

2506.99500144.69 

CIRCUMFERENCES    AND    AREAS   OP  CIRCLES.       119 


Diam 

Circum. 

Area. 

Diam.  I  Circum. 

Area. 

Diam 

Circum.  1  Area. 

799 

2510.13 

501398.97 

867 

2723.76 

590375.16 

935 

2937.39 

686614.71 

8OO 

2513.27 

502654.82 

868 

2726.90 

591737.83 

936 

2940.53 

688084.  19 

801 

2516.42 

503912.25 

869 

2730.04 

593102.06 

937 

2943.67 

689555.24 

802 

2519.56 

505171.24 

870 

2733.19 

594467.87 

938 

2946.81 

691027.86 

803 

2522.70 

50643  1  .  80 

871 

2736.33 

595835.25 

939 

2949.96 

692502  05 

804 

2525.84 

507693.94 

872 

2739.47 

597204.20 

940 

2953.10 

693977.82 

805 

2528.98 

508957.64 

873 

2742.61 

598574.72 

941 

2956.24 

695455.  15 

806 

2532.12 

510222.92 

874 

2745.75 

599946.81 

942 

2959.38 

696934.06 

807 

2535.27 

511489.77 

875 

2748.89 

601320.47 

943 

2962.52 

698414.53 

808 

2538.41 

512758.19 

876 

2752.04 

602695.70 

944 

2965  .  66 

699896.58 

809 

2541.55 

514028.18 

877 

2755.18 

604072.50 

945 

2968.81 

701380.19 

810 

2544.69 

515299.74 

878 

2758.321605450.88 

946 

2971.95 

702865  38 

811 

2547.83 

516572.87 

879 

2761.46 

606830.82 

947 

2975.09 

704352.14 

812 

2550.97 

517847.57 

88O 

2764.60 

608212.34 

948 

2978.23 

705840  47 

813 

2554.11 

519123.84 

881 

27  '67.7  'A 

609595.42 

949 

2981.37 

707330  37 

814 

2557.26 

520401.68 

882 

2770.88 

610980.08 

05O 

2984.51 

708821.84 

815 

2560.40 

521681.10 

883 

2774.03 

612366.31 

951 

2987.65 

7  1  03  1  4  .  88 

816 

2563.54 

522962.08 

884 

2777.  17 

613754.11 

952 

2990.80 

711809.50 

817 

2566.68 

524244.63 

885 

2780.31 

615143.48 

953 

2993  .  94 

713305.68 

818 

2569.82 

525528.76 

886 

2783.45 

616534.42 

954 

2997.08 

714803.43 

819 

2572,96 

526814.46 

887 

2786.59 

617926.93 

955 

3000.22 

716302.76 

820 

2576.11 

528101.73 

888 

2789.73 

619321.01 

956 

3003.36 

717803.66 

821 

2579.25 

529390.56 

889 

2792.88 

620716.66 

957 

3006.50 

719306.12 

822 

2582.39 

530680.97 

890 

2796.02 

622113.89 

958 

3009.65 

720810.16 

823 

2585.53 

531972.95 

891 

2799.16 

623512.68 

959 

3012.79 

722315.77 

824 

2588.67 

533266.50 

892 

2802.30 

624913.04 

960 

3015.93 

723822.95 

825 

2591.81 

534561.62 

893 

2805.44 

626314.98 

961 

3019.07 

725331.70 

826 

2594.96 

535858.32 

894 

2808.58 

627718.49 

962 

3022.21 

726842.02 

827 

2598.10 

537156.58 

895 

2811.73 

629123.56 

963 

3025.35 

728353.91 

828 

2601.24 

538456.41 

896 

2814.87 

630530.21 

964 

3028.50 

729867.37 

829 

2604.38 

539757.82 

897 

2818.01 

631938.43 

965 

3031.64 

731382.40 

830 

2607.52 

541060.79 

898 

2821.15 

633348.22 

966 

3034.78 

732899.01 

831 

2610.66 

542365.34 

899 

2824.29 

634759.58 

967 

3037.92 

734417.18 

832 

2613.81 

543671.46 

900 

2827.43 

636172.51 

968 

3041.06 

735936.93 

833 

2616.95 

544979.15 

901 

2830.58 

637587.01 

969 

3044.20 

737458.24 

834 

2620.09 

546288.40 

902 

2833.72 

639003.09 

970 

3047.34 

738981.13 

835 

2623.23 

547599.23 

903 

2836.86 

640420.73 

971 

3050.49 

740505.59 

836 

2626.37 

548911.63 

904 

2840.00 

641839.95 

972 

3053.63 

74203  1  .  62 

837 

2629.51 

550225.61 

905 

2843.14 

643260.73 

973 

3056.77 

743559.22 

838 

2632.65 

551541.15 

906 

2846.28 

644683  .  09 

974 

3059.91 

745088.39 

839 

2635.80 

552858.26 

907 

2849.42 

646107.01 

975 

3063.05 

746619.  13 

840 

2638.94 

554176.94 

908 

2852.57 

647532.51 

976 

3066.19 

748151.44 

841 

2642.08 

555497.20 

909 

2855.71 

648959.58 

977 

3069.34 

749685.32 

842 

2645.22 

556819.02 

910 

2858.85 

650388.22 

978 

3072.48 

751220.78 

843 

2648.36 

558142.42 

911 

2861.99 

651818.43 

979 

3075.62 

752757.80 

844 

2651.50 

559467.39 

912 

2865.13 

653250.21 

98O 

3078.76 

754296.40 

845 

2654.65 

560793.92 

913 

2868.27 

654683.56 

981 

3081.90 

755836.56 

846 

2657.79 

562122.03 

914 

2871.42 

656118.48 

982 

3085.04 

757378.30 

847 

2660.93 

563451.71 

915 

2874.56 

657554.98 

983 

3088.19 

758921.61 

848 

2664.07 

564782.96 

916 

2877.70 

658993  .  04 

984 

3091  .33 

760466.48 

849 

2667.21 

566115.78 

917 

2880.84 

660432.68 

985 

3094.47 

762012.93 

850 

2670.35 

567450.17 

918 

2883.98 

661873.88 

986 

3097.61 

763560.95 

851 

2673.50 

568786.14 

919 

2887.12 

663316.66 

987 

3100.75 

765110.54 

852 

2676.64 

570123.67 

92O 

2890.27 

664761.01 

988 

3103.89 

766661.70 

853 

2679.78 

571462.77 

921 

2893.41 

666206.92 

989 

3107.04 

768214.44 

854 

2682.92 

572803.45 

922 

2896.55 

667654.41 

99O 

3110.18 

769768.74 

855 

2686.06 

574145.69 

923 

2899.69 

669103.47 

991 

3113.32 

771324.61 

856 

2689.20 

575489.51 

924 

2902.83 

670554.10 

992 

3116.46 

772882.06 

857 

2692.34 

576834.90 

925 

2905.97 

672006.30 

993 

3119.60 

774441.07 

858 

2695.49 

578181.85 

926 

2909.11 

673460.08 

994 

3122.74 

776001.66. 

859 

2698.63 

579530.38 

927 

2912.26 

674915.42 

995 

3125.88 

777563.82 

860 
861 

2701.77 
2704.91 

580880.48 
582232.15 

928 
929 

2915.40676372.33 
2918.54  677830.82 

996 
997 

3129.03 
3132.17 

779127.54 
780692  84 

862 

2708.05 

583585.39 

930 

2921.68:679290.87 

998 

3135.31 

782259.7? 

863 

2711.19 

584940.20 

931 

2924.82  680752.50 

999 

3138.45 

783828  15 

864 

2714.34 

586296.59 

932 

2927.96682215.69 

1000 

3141.59 

785398.16 

865 

2717.48 

587654.54 

933 

2931  .  11  683680.46 

866 

m 

2720.62 

589014.07 

934 

2934.25  685146.80 

120     CIRCUMFERENCE  OF  CIRCLES,  FEET  AND  INCHED 


—  ^OG  —  <*JiN.OenNOOe^\NOONcNmoo^~moo  —  ^  t>.  o  m  r^  o  m 


—  —  —  CN  CN  «N  en 


•1-— OOONooh-.NOm-'^-eneN  —  OONOOooi^NOin-feneN  —  o  ON~OO  r  „  -  , 


rNNOenot^'3-  — ooineNC 


V  in  t>.'  a^  —  o  CN 
•CNmoo  —  •^•oo  —  ^4 


rNOoeN-<rmi>.ONOOcNcnmt>.ONOc 


- 

^B  —  oo  in  CN  ON  vo  co  o  r>.  •*  —  oo  in  C 


^jesmoo  —  •<j-f>.O'<j-t>.oen\ooNeNNOONeNinoo  —  -<roo  — •«j-h>.oenNooenNOONeN 
p^  mmmi  r>,r>,      oo      ONONQNONO 


r;oNONoooooooooooooooooooooor>.r>«rs«.r>^r>.t>.r>»r>»r<«t>«rs».NONONONONONONONONO 

£^*^3^*^w^^^*^or^^"«^«N^^f<\o^^^^^^^Ndf^ 


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m  in  o  vo  so  i>«  t>«  t>«  oo  oo  oo  oo  ON  ON  ON 


N  —  ooNoooor>«NO>n^tTncN  —  ON 

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ocnr>iOfnNOONCNinoNCNmoo  —  -*r>.o-^-t>.oenvOONCNvOONCNino    — 
—  —  —  CN  CN  CN  CN  en  en  en  -«r  if  -^-  m  m  m  NO  NO  NO  t>.  i>.  t>«  r>.  oo  oo  oo  ON  ON  ON  o 


<  — 

ONONONONOu^u 
O  en  o  r^  ^  •—  o 

'    '        * 


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-—  oo  in  CN 

* 


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o  —  moo  —  -<i 
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ooin<NONNOenor>.-«r  — 
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—  —  —  CN  CN  CN  en  en  en  -<r  ^n-  >n  in  m  in  NO  NO  NO  t>.  t>.  t>,  oo  oo  oo  ON  ON  ON  o 


AREAS  OF  THE  SEGMENTS  OF  A  CIRCLE. 


121 


AREAS  OF  THE   SEGMENTS  OF  A  CIRCLE. 

(Diameter=l;  Rise  or  Height  in  parts  of  Diameter  being  given.) 

RULE  FOR  USE  OF  THE  TABLE. — Divide  the  rise  or  height  of  the  segment 
by  the  diameter.  Multiply  the  area  in  the  table  corresponding  to  the 
quotient  thus  found  by  the  square  of  the  diameter. 

//  the  segment  exceeds  a  semicircle  its  area  is  area  of  circle  —  area  of  seg- 
ment whose  rise  is  (diarn.  of  circle  —  rise  of  given  segment). 

Given  chord  and  rise,  to  find  diameter.     Diam.  =  (square  of  half  chord -*- 
rise)  +  rise.    The  half  chord  is  a  mean  proportional  between  the  two  parts  . 
into  which  the  chord  divides  the  diameter  which  is  perpendicular  to  it. 


Rise 
-f- 

Diam. 

Area. 

Rise 
^ 
Diam. 

Area. 

Rise 
Diam. 

Area. 

Rise 
Diam. 

Area. 

Rise 
-i- 

Diam. 

Area. 

.001 

.00004 

.054 

•01646 

.107 

.04514 

.16 

.08111 

.213 

.12235 

.002 

.00012 

.055 

.01691 

.108 

.04576 

.161 

.08185 

.214 

.12317 

.003 

.00022 

.056 

.01737 

.109 

.04638 

.162 

.08258 

.215 

.12399 

.004 

.00034 

.057 

.01783 

.11 

.04701 

.163 

.08332 

.216 

.12481 

.005 

.00047 

.058 

.01830 

.111 

.04763 

.164 

.08406 

.217 

.12563 

.006 

.00062 

.059 

.01877 

.112 

.04826 

.165 

.08480 

.218 

.12646 

.007 

.00078 

.06 

.01924 

.113 

.04889 

.166 

.08554 

.219 

.12729 

.008 

.00095 

.061' 

.01972 

.114 

.04953 

.167 

.08629 

.22 

.12811 

.009 

.00113 

.062 

.02020 

.115 

.05016 

.168 

.08704 

.221 

.12894 

.01 

.00133 

.063 

.02068 

.116 

.05080 

.169 

.08779 

.222 

.12977 

.011 

.00153 

.064 

.02117 

.117 

.05145 

.17 

.08854 

.223 

.13060 

.012 

.00175 

.065 

.02166 

.118 

.05209 

.171 

.08929 

.224 

.13144 

.013 

.00197 

.066 

.02215 

.119 

.05274 

.172 

.09004 

.225 

.13227 

.014 

.0022 

.067 

.02265 

.12 

.05338 

.173 

.09080 

.226 

.13311 

.015 

.00244 

.068 

.02315 

.121 

.05404 

.174 

.09155 

.227 

.13395 

.016 

.00268 

.069 

.02366 

.122 

.05469 

.175 

.0923  1 

.228 

.13478 

.017 

.00294 

.07 

.02417 

.123 

.05535 

.176 

.09307 

.229 

.13562 

.018 

.0032 

.071 

.02468 

.124 

.05600 

.177 

.09384 

.23 

.13646 

.019 

.00347 

.072 

.02520 

.125 

.05666 

.178 

.09460 

.231 

.13731 

.02 

.00375 

.073 

.02571 

.126 

.05733 

.179 

.09j37 

.232 

.13815 

.021 

.00403 

.074 

.02624 

.127 

.05799 

.18 

.09613 

.233 

.13900 

.022 

00432 

.075 

.02676 

.128 

.05866 

.181 

.09690 

.234 

.13984 

.023 

.00462 

.076 

.02729 

.129 

.05933 

.182 

.09767 

.235 

.14069 

.024 

.00492 

.077 

.02782 

.13 

.06000 

.183 

.09845 

.236 

.14154 

.025 

.00523 

.078 

.02836 

.131 

.06067 

.184 

.09922 

.237 

.14239 

.026 

.00555 

.079 

.02889 

.132 

.06135 

.185 

.10000 

.238 

.14324 

.027 

.00587 

.08 

.02943 

.133 

.06203 

.186 

.10077 

.239 

.14409 

.028 

.00619 

.081 

.02998 

.134 

.06271 

.187 

.10155 

.24 

.14494 

,029 

.00653 

.082 

.03053 

.135 

.06339 

.188 

.10233 

.241 

.14580 

.03 

.00687 

.083 

.03108 

.136 

.06407 

.189 

.10312 

.242 

.14666 

.031 

.00721 

.084 

.03163 

.137 

.06476 

.19 

.10390 

.243 

.14751 

.032 

.00756 

.085 

.03219 

.138 

.06545 

.191 

.10469 

.244 

.14837 

.033 

.00791 

.086 

.03275 

.139 

.06614 

.192 

.10547 

.245 

.14923 

.034 

.00827 

.087 

.03331 

.14 

.06683 

.193 

.10626 

.246 

.15009 

.035 

.00864- 

.088 

.03387 

.141 

.06753 

.194 

.10705 

.247 

.15095 

.036 

.00901 

.089 

.03444 

.142 

.06822 

.195 

.10784 

.248 

.15182 

.037 

.00938 

.09 

.03501 

.143 

.06892 

.196 

.10864 

.249 

.15268 

.038 

.00976 

091 

.03559 

.144 

.06963 

.197 

.10943 

.25 

.15355 

.039 

.01015 

.092 

.03616 

.145 

.07033 

.198 

.11023 

.251 

.15441 

.04 

.01054 

.093 

.03674 

.146 

.07103 

.199 

.11102 

.252 

.15528 

.041 

.01093 

.094 

.03732 

.147 

.07174 

.2 

.11182 

.253 

.15615 

.042 

.01133 

.095 

.03791 

.148 

.07245 

.201 

.11262 

.254 

.15702 

.043 

.01173 

.096 

.03850 

.149 

.07316 

.202 

.11343 

.255 

.15789 

.044 

.01214 

.097 

.03909 

.15 

.07387 

.203 

.11423 

.256 

.15876 

.045 

.01255 

.098 

.03968 

.151 

.07459 

.204 

.11504 

.257 

.15964 

.046 

.01297 

.099 

.04028 

.152 

.07531 

.205 

.11584 

.258 

.16051 

.047 

.01339 

| 

.04087 

.153 

.07603 

.206 

.11665 

.259 

.16139 

.048 

.01382 

J01 

.04148 

.154 

.07675 

.207 

.  1  1  746 

.26 

.16226 

.049 

.01425 

.102 

.04208 

.155 

.07747 

.208 

.11827 

.261 

.16314 

.05 

.01468 

.103 

.04269 

.156 

.07819 

.209 

.11908 

.262 

.16402 

.051 

.01512 

.104 

.04330 

.157 

.07892 

.21 

.11990 

.263 

.16490 

.052 

.01556 

.105 

.04391 

.158 

.07965 

.211 

.12071 

.264 

.16578 

.053 

.01601 

.106 

.04452 

.159 

.08038 

.212 

.12153 

.265 

.16666 

122 


MATHEMATICAL  TABLES, 


Rise 

•4- 

Diam. 

Area. 

Rise 
Diam. 

Area. 

Rise 

•*- 
Diam. 

Area. 

Rise 

•*• 
Diana. 

Area. 

Rise 

•*• 
Diam. 

Area. 

.266 

.16755 

.313 

.21015 

.36 

.25455 

.407 

.30024 

.454 

.34676 

.267 

.16643 

.314 

.21108 

.361 

.25551 

.408 

.30122 

.455 

-.34776 

.268 

.16932 

.315 

.21201 

.362 

.25647 

.409 

.30220 

.456 

.34876 

.269 

17020 

316 

.21294 

.363 

.25743 

.41 

.30319 

.457 

.34975 

.27 

.17109 

.317 

.21387 

.364 

.25839 

.411 

.30417 

.458 

.35075 

.271 

.17198 

.318 

.21480 

.365 

.25936 

.412 

.30516 

.459 

.35175 

.272 

.17287 

.319 

.21573 

.366 

.26032 

.413 

.30614 

.46 

.35274 

.273 

.17376 

.32 

.21667 

.367 

.26128 

.414 

.30712 

.461 

.35374 

.274 

.17465 

.321 

.21760 

.368 

.26225 

.415 

.30811 

.462 

.35474 

.275 

.17554 

.322 

.21853 

.369 

.26321 

.416 

.30910 

.463 

.35573 

.276 

.17644 

.323 

,21947 

.37 

.26418 

.417 

.3  1  008 

.464 

.35673 

.277 

.17733 

.324 

.22040 

.37! 

.265  1  4 

.418 

.31107 

.465 

.35773 

.278 

.17823 

.325 

.22134 

.372 

.26611 

.419 

.31205 

.466 

.35873 

.279 

.17912 

.326 

.22228 

.373 

.26708 

.42 

.31304 

.467 

.35972 

.28 

.18002 

.327 

.22322 

.374 

.26805 

.421 

.31403 

.468 

.36072 

.281 

.18092 

.328 

.22415 

.375 

.26901 

.422 

.31502 

.469 

.36172 

.282 

.18182 

.329 

.22509 

.376 

.26998 

.423 

.31600 

.47 

.36272 

.283 

.18272 

.33 

.22603 

.377 

.27095 

.424 

.31699 

.471 

.36372 

.284 

.18362 

.331 

.22697 

.378 

.27192 

.425 

.3  1  798 

.472 

.36471 

.285 

.18452 

.332 

.22792 

.379 

.27289 

.426 

.31897 

.473 

.36571 

.286 

.18542 

.333 

.22886 

.38 

.27386 

.427 

.31996 

.474 

.36671 

.287 

.18633 

.334 

.22980 

.381 

.27483 

.428 

.32095 

.475 

.36771 

.288 

.18723 

.335 

.23074 

.382 

.27580 

.429 

.32194 

.476 

.36871 

.289 

.18814 

.336 

.23169 

.383 

.27678 

.43 

.32293 

.477 

.36971 

.29 

.18905 

.337 

.23263 

.384 

.27775 

.431 

.32392 

.478 

.37071 

.291 

.18996 

.338 

.23358 

.385 

.27872 

.432 

.32491 

.479 

.37171 

.292 

.19086 

.339 

.23453 

.386 

.27969 

.433 

.32590 

.48 

.37270 

293 

.19177 

.34 

.23547 

.387 

.28067 

.434 

.32689 

.481 

.37370 

294 

.19268 

.341 

.23642 

.388 

.28164 

.435 

.32788 

.482 

.37470 

.295 

.19360 

.342 

.23737 

.389 

.28262 

.436 

.32887 

.483 

.37570 

.296 

.19451 

.343 

.23832 

.39 

.28359 

.437 

.32987 

.484 

.37670 

.297 

.19542 

.344 

.23927 

.391 

.28457 

.438 

.33086 

.485 

.37770 

.298 

.19634 

.345 

.24022 

.392 

.28554 

.439 

.33185 

.486 

.37870 

.299 

.19725 

.346 

.24117 

.393 

.28652 

.44 

.33284 

.487 

.37970 

.3 

.19817 

.347 

.24212 

.394 

.28750 

.441 

.33384 

.488 

.38070 

.301 

.19908 

.348 

.24307 

.395 

.28848 

.442 

.33483 

.489 

.38170 

.302 

.20000 

.349 

.24403 

.396 

.28945 

.443 

.33582 

.49 

.38270 

.303 

.20092 

.35 

.24498 

.397 

.29043 

.444 

.33682 

.491 

.38370 

,304 

.20184 

.351 

.24593 

.398 

.29141 

.445 

.33781 

.492 

.38470 

.305 

.20276 

.352 

.24689 

.399 

.29239 

.446 

.33880 

.493 

.38570 

.306 

.20368 

.353 

.24784 

.4 

.29337 

.447 

.33980 

.494 

.38670 

.307 

.20460 

.354 

.24880 

".401 

.29435 

.448 

.34079 

.495 

.38770 

.308 

.20553 

.355 

.24976 

.402 

.29533 

.449 

.34179 

.496 

.38870 

.309 

.20645 

.356 

.25071 

.403 

.29631 

.45 

.34278 

.497 

.38970 

.31 

.20738 

.357 

.25167 

.404 

.29729 

.451 

.34378 

.498 

.39070 

.311 

.20830 

.358 

.25263 

.405 

.29827 

.452 

.34477 

.499 

.39170 

.312 

.20923 

.359 

.25359 

.406 

.29926 

.453 

.34577 

.5 

.39270 

For  rules  for  finding  the  area  of  a  segment  see  Mensuration,  page  60. 

LENGTHS  OF  CIRCULAR  ARCS. 

(Degrees  being  given.     Radius  of  Circle  =  1.) 

FORMULA.  —  Length  of  arc  =          g          X  radius  X  number  of  degrees. 

RULE.  —  Multiply  the  factor  in  the  table  (see  next  page)  for  any  given 
number  of  degrees  by  the  radius. 
EXAMPLE.  —  Given  a  curve  of  a  radius  of  55  feet  and  an  angle  of  7  8°  20'. 

r  Factor  from  table  for  78° 1 .3613568 

Factor  from  table  for  20' .0058178 

Factor. 1.3671740 

1.3671746X55  = 


LENGTHS   OP  CIRCULAR  ARCS. 


FACTORS  FOR  LENGTHS  OF  CIRCULAR  ARCS. 


123 


Degrees. 

Minutes. 

1 

.0174533 
.0349066 

61 
62 

1  .0646508 
1.0821041 

121 
122 

2.1118484 
2.1293017 

1 
2 

.0002909 
.0005818 

3 

.0523599 

63 

1.0995574 

123 

2.1467550 

3 

.0008727 

A 

.0698132 

64 

1.1170107 

124 

2.1642083 

4 

.0011636 

5 

.0872665 

65 

1.1344640 

125 

2.1816616 

5 

.0014544 

6 

.1047198 

66 

1.1519173 

126 

2.1991149 

6 

.0017453 

7 

.1221730 

67 

1.1693706 

127 

2.2165682 

7 

.0020362 

8 

.1396263 

68 

1.1868239 

128 

2.2340214 

8 

.0023271 

9 

.1570796 

69 

1.2042772 

129 

2.2514747 

9 

.0026  1  80 

10 

.1745329 

70 

1.2217305 

130 

2.2689280 

10 

.0029089 

11 

.1919862 

71 

1.2391838 

131 

2.2863813 

11 

.0031998 

12 

.2094395 

72 

1.2566371 

132 

2.3038346 

12 

.0034907 

13 

.2268928 

73 

1  .2740904 

133 

2.3212879 

13 

.0037815 

14 

.2443461 

74 

1.2915436 

134 

2.3387412 

14 

.0040724 

15 

.2617994 

75 

1.3089969 

135 

2.3561945 

15 

.0043633 

16 

.2792527 

'  76 

1.3264502 

136 

2.3736478 

16 

.0046542 

17 

.2967060 

77 

1.3439035 

137 

2.3911011 

17 

.0049451 

18 

.3141593 

78 

1.3613568 

138 

2.4085544 

18 

.0052360 

19 

.3316126 

79 

1.3788101 

139 

2.4260077 

19 

.0055269 

20 

.3490659 

80 

1.3962634 

140 

2.4434610 

20 

.0058178 

21 

.3665191 

81 

1.4137167 

141 

2.4609142 

21 

.0061087 

22 

.3839724 

82 

1  .43  1  1  700 

142 

2.4783675 

22 

.0063995 

23 

.4014257 

83 

1  .4486233 

143 

2.4958208 

23 

.0066904 

24 

.4188790 

84 

1  .4660766 

144 

2.5132741 

24 

.0069813 

25 

.4363323 

85 

1.4835299 

145 

2.5307274 

25 

.0072722 

26 

.4537856 

86 

1.5009832 

146 

2.5481807 

26 

.0075631 

27 

.4712389 

87 

1.5184364 

147 

2.5656340 

27 

.007854(1 

28 

.4886922 

88 

1.5358897 

148 

2.5830873 

28 

.0081449 

29 

.5061455 

89 

1.5533430 

149 

2.6005406 

29 

.0084358 

30 

.5235988 

90 

1.5707963 

150 

2.6179939 

30 

.008726(1 

31 

.5410521 

91 

1  .5882496 

151 

2.6354472 

31 

.0090175 

32 

.5585054 

92 

1.6057029 

152 

2.6529005 

32 

.0093084 

33 

.5759587 

93 

1.6231562 

153 

2.6703538 

33 

.0095993 

34 

.5934119 

94 

1  .6406095 

154 

2.6878070 

34 

.0098902 

35 

.6108652 

95 

1.6580628 

155 

2.7052603 

35 

.0101811 

36 

.6283185 

96 

1.6755161 

156 

2.7227136 

36 

.0104720 

37 

.6457718 

97 

1  .6929694 

157 

2.7401669 

37 

.0107629 

38 

.6632251 

98 

1.7104227 

158 

2.7576202 

38 

.0110538 

39 

.6806784 

99 

1.7278760 

159 

2.7750735 

39 

.0113446 

40 

.6981317 

100 

1.7453293 

160 

2.7925268 

40 

.0116355 

41 

.7155850 

101 

1.7627825 

161 

2.8099801 

41 

.0119264 

42 

.7330383 

102 

1.7802358 

162 

2.8274334 

42 

.0122173 

43 

.7504916 

103 

1.7976891 

163 

2.8448867 

43 

.0125082 

44 

.7679449 

104 

1.8151424 

164 

2.8623400 

44 

.0127991 

45 

.7853982 

105 

1.8325957 

165 

2.8797933 

45 

.0130900 

46 

.8028515 

106 

1  .8500490 

166 

"2.8972466 

46 

.0133809 

47 

.8203047 

107 

1.8675023 

167 

2.9146999 

47 

.0136717 

48 

.8377580 

108 

1.8849556 

168 

2.9321531 

48 

.0139626 

49 

.8552113 

109 

1  .9024089 

169 

2.9496064 

49 

.0142535 

50 

.8726646 

110 

1.9198622 

170 

2.9670597 

50 

.0145444 

51 

.8901179 

111 

1.9373155 

171 

2.9845130 

51 

.0148353 

52 

.9075712 

112 

1.9547688 

172 

3.0019663 

52 

.0151262 

53 

.9250245 

113 

1  19722221 

173 

3.0194196 

53 

.0154171 

54 

.9424778 

114 

1  .9896753 

174 

3.0368729 

54 

.0157080 

55 

.9599311 

115 

2.0071286 

175 

3.0543262 

55 

.0159989 

56 

.9773844 

116 

2.0245819 

176 

3.0717795 

56 

.0162897 

57 

.9948377 

117 

2.0420352 

177 

3.0892328 

57 

.0165806 

58 

1.0122910 

118 

2.0594885 

178 

3.1066861 

58 

.0168715 

59 

1.0297443 

119 

2.0769418 

179 

3.1241394 

59 

.0171624 

60 

1.0471976 

120 

2.0943951 

180 

3.1415927 

60 

.0174533 

124 


MATHEMATICAL 


LENGTHS  Off  CtHCtJLAll  ARCS. 

(Diameter  =-  1.    Given  the  Chord  and  Height  of  the  Arc.) 

RULE  FOR  USE  OF  THE  TABLE.  —  Divide  the  height  by  the  chord.  Find 
In  the  column  of  heights  the  number  equal  to  this  quotient.  Take  out  the 
corresponding  number  from  the  column  of  lengths.  Multiply  this  last 
number  by  the  length  of  the  given  chord;  the  product  will  be  length  of  the 

If  the  arc  is  greater  than  a  semicircle,  first  find  the  diameter  from  the 
formula,  Diam.  =  (square  of  half  chord  -f-  rise)  +  rise;  the  formula  is  true 
whether  the  arc  exceeds  a  semicircle  or  not.  Then  find  the  circumference. 
From  the  diameter  subtract  the  given  height  of  arc,  the  remainder  will  be 
height  of  the  smaller  arc  of  the  circle;  find  its  length  according  to  the  rule, 
and  subtract  it  from  the  circumference. 


Hgts. 

Lgths. 

Hgts. 

Lgths. 

Hgts. 

Lgths. 

Hgts. 

Lgths. 

Hgts. 

Lgths. 

0.001 

.00002 

0.15 

.05896 

0.238 

.14480 

0.326 

.26288 

0.414 

.40788 

.005 

.00007 

.152 

.06051 

.24 

.14714 

.328 

.26588 

.416 

.41145 

.01 

.00027 

.154 

.06209 

.242 

.14951 

.33 

.26892 

.418 

.41503 

.015 

.00061 

.156 

.06368 

.244 

.15189 

.332 

.27196 

.42 

.41861 

.02 

.00107 

.158 

.06530 

.246 

.15428 

.334 

.27502 

.422 

.42221 

.025 

.00167 

.16 

.06693 

.248 

.15670 

.336 

.27810 

.424 

.42583 

.03 

.00240 

.162 

.06858 

.25 

.15912 

.338 

.28118 

.426 

.42945 

.035 

.00327 

.164 

.07025 

.252 

.16156 

.34 

.28428 

.428 

.43309 

.04 

.00426 

.166 

.07194 

.254 

.16402 

:342 

.28739 

.43 

.43673 

.045 

.00539 

.168 

.07365 

.256 

.16650 

.344 

.29052 

.432 

.44039 

.05 

.00665 

.17 

.07537 

.258 

.16899 

.346 

.29366 

.434 

.44405 

.055 

.00805 

.172 

.07711 

.26 

.17150 

.348 

.29681 

.436 

.44773 

.06 

.00957 

.174 

.07888 

.262 

.17403 

.35 

.29997 

.438 

.45142 

.065 

.01123 

.176 

.08066 

.264 

.17657 

.352 

.30315 

.44 

.45512 

.07 

.01302 

.178 

.08246 

.266 

.17912 

.354 

.30634 

.442 

.45883 

.075 

.01493 

.18 

.08428 

.268 

.18169 

.356 

.30954 

.444 

.46255 

.08 

.01698 

.182 

.08611 

.27 

.18429 

.358 

.31276 

.446 

.46628 

.085 

.01916 

.184 

.08797 

.272 

.18689 

.36 

.31599 

.448 

.47002 

.09 

.02146 

.186 

.08984 

.274 

.18951 

.362 

.31923 

.45 

.47377 

.095 

.02389 

.188 

.09174 

.276 

.19214 

.364 

.32249 

.452 

.47753 

.10 

.02646 

.19 

.09365 

.278 

.19479 

.366 

.32577 

.454 

.48131 

.102 

.02752 

.192 

.09557 

.28 

.19746 

.368 

.32905 

.456 

.48509 

.104 

.02860 

.194 

.09752 

.282 

.20014 

.37 

.33234 

.458 

.48889 

.106 

.02970 

.196 

.09949 

.284 

.20284 

.372 

.33564 

.46 

.49269 

.108 

.03082 

.198 

.10147 

.286 

.20555 

.374 

.33896 

.462 

.49651 

.11 

.03196 

.20 

.10347 

.288 

.20827 

.376 

.34229 

.464 

.50033 

.112 

.03312 

.202 

.10548 

.29 

.21102 

.378 

.34563 

.466 

.50416 

.114 

.03430 

.204 

.10752 

.292 

.21377 

.38 

.34899 

.468 

.50800 

.116 

.03551 

.206 

.10958 

.294 

.21654 

.382 

.35237 

.47 

.51185 

.118 

.03672 

.208 

.11165 

.296 

.21933 

.384 

.35575 

.472 

.51571 

.12 

.03797 

.21 

.11374 

.298 

.22213 

.386 

.35914 

.474 

.51958 

.122 

.03923 

.212 

.11584 

.30 

.22495 

.388 

.36254 

.476 

.52346 

.124 

.04051 

.214 

.11796 

.302 

.22778 

.39 

.36596 

.478 

.52736 

.126 

.04181 

.216 

.12011 

.304 

.23063 

.392 

.36939 

.48 

.53126 

.128 

.04313 

.218 

.12225 

.306 

.23349 

.394 

.37283 

.482 

.53518 

.13 

.04447 

.22 

.12444 

.308 

.23636 

.396 

.37628 

.484 

.53910 

.132 

.04584 

.222 

.12664 

.31 

.23926 

.398 

.37974 

.486 

.54302 

.134 

.04722 

.224 

.12885 

.312 

.24216 

>  .40 

.38322 

.488 

.54696 

.136 

.04862 

.226 

.13108 

.314 

.24507 

.402 

.38671 

.49 

.55091 

.138 

.05003 

.228 

.13331 

.316 

.24801 

.404 

.39021 

.492 

.55487 

.14 

.05147 

.23 

.13557 

.318 

.25095 

.406 

.39372 

.494 

.55854 

.142 

.05293 

.232 

.13785 

.32 

.25391 

.408 

.39724 

.496 

.56282 

.144 

.05441 

.234 

.14015 

.322 

.25689 

.41 

.40077 

.498 

.56681 

.146 

.05591 

.236 

.14247 

.324 

.25988 

.412 

.40432 

.50 

.57080 

.148 

.05743 

CIRCLES  AND  SQUARES   OF   EQUAL  AREA.         125 


Diameters  of  Circles  and  Sides  of  Squares  of  Same  Area. 

Diameter  of  circle  *  1.128379  X  side  of  square  of  same  area. 
Side  of  square        «  0.886227  X  diameter  of  circle  of  same  area. 


Diam.  of  Cir- 
cle or  Side 
of  Square. 

Side  of 
Square 
Equiva- 
lent to 
Circle. 

Diam.  of 
Circle 
Equiva- 
lent to 
Square. 

Diam.  of  Cir- 
cle or  Side 
of  Square. 

Side  of 
Square 
Equiva- 
lent to 
Circle. 

Diam.  of 
Circle 
Equiva- 
lent to 
Square. 

Diam.  of  Cir- 
cle or  Side 
of  Square. 

Side  of 
Square 
Equiva- 
lent to 
Circle. 

Diam.  of 
Circle 
Equiva- 
lent to 
Square. 

1 

0.886 

1  .128 

34 

30.132 

38.365 

67 

59.377 

75.601 

2 

1.772 

2.257 

35 

31.018 

39.493 

68 

60.263 

76.730 

3 

2.659 

3.385 

36 

31.904 

40.622 

69 

61  .150 

77.858 

4 

3.545 

4.514 

37 

32.790 

41.750 

70 

62  .  036 

78.987 

5 

4.431 

5.642 

38 

33.677 

42.878 

71 

62.922 

80.115 

6 

5.317 

6.770 

39 

34.563 

44.007 

72 

63.808 

81.243 

7 

6.204 

7.899 

40 

35.449 

45.135 

73 

64.695 

82.372 

8 

7.090 

9.027 

41 

36.335 

46  .  264 

74 

65.581 

83.500 

9 

7.976 

10.155 

42 

37.222 

47.392 

75 

66.467 

84.628 

10 

8.862 

11.284 

43 

38.108 

48.520 

76 

67.353 

85.757 

11 

9.748 

12.412 

44 

38.994 

49.649 

77 

68.239 

86.885 

12 

10.635 

13.541 

45 

39.880 

50.777 

78 

69.126 

tttt.014 

13 

11.521 

14.669 

46 

40.766 

51  .905 

79 

70.012 

89.142 

14 

12.407 

15.797 

47 

41  .653 

53.034 

80 

70.898 

90.270 

15 

13.293 

16.926 

48 

42.539 

54.162 

81 

71  .784 

91  .399 

16 

14.180 

18.054 

49 

43.425 

55.291 

82 

72.671 

92.527 

17 

15.066 

19.182 

50 

44.311 

56.419 

83 

73.557 

93.655 

18 

15.952 

20.311 

51 

45.198 

57.547 

84 

74.443 

94.784 

19 

16.838 

21.439 

52 

46.084 

58.676 

85 

75.330 

95.912 

20 

17.725 

22.568 

53 

46.970 

59.804 

86 

76.216 

97.041 

21 

18.611 

23.696 

54 

47.856 

60.932 

87 

77.102 

98.169 

22 

19.497 

24.824 

55 

48.742 

62.061 

88 

77.988 

99.297 

23 

20.383 

25.953 

56 

49.629 

63.189 

89 

>8.874 

100.426 

24 

21.269 

27.081 

57 

50.515 

64.318 

90 

79.760 

101  .554 

25 

22.156 

28.209 

58 

51.401 

65.446 

91 

80.647 

102.682 

26 

23.042 

29.338 

59 

52.287 

66.574 

92 

81.533 

103.811 

27 

23.928 

30.466 

60 

53.174 

67.703 

93 

82.419 

104.939 

28 

24.814 

31.595 

61 

54.060 

68.831 

94 

83.305 

106.068 

29 

25.701 

32.723 

62 

54.946 

69.959 

95 

84  192 

107.196 

30 

26.587 

33.851 

63 

55.832 

71  .088 

96 

85.078 

108.324 

31 

27.473 

34.980 

64 

56  719 

72.216 

97 

85.964 

109.453 

32 

28.359 

36.108 

65 

57.605 

73.345 

98 

86.850 

110.581 

33 

29.245 

37.237 

66 

58.491 

74.473 

99 

87.736 

111.710 

Number  of  Circles  that  can  be  Inscribed  within  a  Larger  Circle.  • 

N  =  Number  of  circles;  D  =  diam.  of  enclosing  circle;  d  =  diam.  of 
inscribed  circles. 

Obtain  the  ratio  of  D  -5-  d  and  find  the  value  nearest  to  it  in  the 
table.  Opposite  this  value  under  Ar,  find  the  number  of  circles  of 
diameter  d  that  can  be  inscribed  in  a  circle  of  diameter  D. 


N 

D/d 

N 

D/d 

N 

D/d 

N 

D/d 

N 

D/d 

N 

D/d 

N 

D/d 

2 

2.00 

13 

4.23 

24 

5.72 

35 

6.86 

46 

7.81 

85 

10.46 

140 

13.26 

3 

2.15 

14 

4.41 

25 

5.81 

36 

7.00 

47 

7.92 

90 

10.73 

145 

13.49 

4 

2.41 

15 

4.55 

26 

5.92 

37 

7.00 

48 

8.00 

95 

11.15 

150 

13.72 

-  5 

2.70 

16 

4.70 

27 

6.00 

38 

7.08 

49 

8.03 

100 

11.34 

155 

13.95 

6 

3.00 

17 

4.86 

28 

6.13 

39 

7.18 

50 

8.13 

105 

11.60 

160 

14.17 

7 

3.00 

18 

5.00 

29 

6.23 

40 

7.31 

55 

8.21 

110 

11.85 

165 

14.39 

8 

3.31 

19 

5.00 

30 

6.40 

41 

7.39 

60 

8.94 

115 

12.10 

170 

14.60 

9 

3.61 

20 

5.18 

31 

6.44 

42 

7.43 

65 

9.25 

120 

12.34 

175 

14.81 

10 

3.80 

21 

5.31 

32 

6.55 

43 

7.61 

70 

9.61 

125 

12.57 

180 

15.01 

11 

3.92 

22 

5.49 

33 

6.70 

44 

7.70 

75 

9.93 

130 

12.80 

185 

15.20 

12 

4.05 

23 

5.61 

34 

6.76 

45 

7.72 

80 

10.20 

135 

13.06 

190 

15.39 

126 


MATHEMATICAL  TABLES. 


SPHERES. 

(Some  errors  of  1  in  the  last  figure  only.    From  TRAUTWINE.) 


Diam 

Sur- 
face. 

Vol- 
ume. 

Diam 

Sur- 
face. 

Vol- 
ume. 

Diam. 

Sur- 
face. 

Vol- 
ume. 

V32 

.0030 

.0000 

31/4 

33.18 

17.97 

97/8 

306.36 

504.21 

Vie 

.0122 

.0001 

5/16 

34.47 

19.03 

10. 

314.16 

523.60 

3/32 

.0276 

.0004 

3/8 

35.78 

20.129 

1/8 

322.06 

543.48 

1/8 

.0490 

.0010 

7/16 

37.122 

21.268 

1/4 

330.06 

563.86 

5/32 

.0767 

.0020 

1/2 

38.484 

22.449 

3/8 

338.16 

584.74 

3/16 

.1104 

.0034 

9/16 

39.872 

23.674 

1/2 

346.36 

606.13 

7/32 

.1503 

.0054 

5/8 

41.283 

24.942 

5/3 

354.66 

628.04 

1/4 

.1963 

.0081 

H/16 

42.719 

26.254 

3/4 

363.05 

650.46 

9/32 

.2485 

.0116 

3/4 

44.179 

27.61 

7/8 

371.54 

673.42 

5/16 

.3068 

.0159 

13/16 

45.664 

29.016 

11. 

380.13 

696.91 

U/32 

.3712 

.0212 

7/8 

47.173 

30.466 

1/8 

388.83 

720.95 

3/8 

.44179 

.0276 

15/16 

48.708 

31.965 

1/4 

397.61 

745.51 

13/32 

.51848 

.0351 

4. 

50.265 

33.510 

3/8 

406.49 

770.64 

7/16 

.60132 

.0438 

1/8 

53.456 

36.751 

V2 

415.48 

796.33 

15/32 

.69028 

.0539 

V4 

56.745 

40.195 

5/8 

424.50 

822.58 

1/2 

.78540 

.0654 

3/8 

60.133 

43.847 

3/4 

433.73 

849.40 

9/16 

.99403 

.0931 

1/2 

63.617 

47.713 

7/8 

443.01 

876.79 

5/8 

1.2272 

.12783 

5/8 

67.201 

51.801 

12. 

452.39 

904.78 

H/16 

1.4849 

.17014 

3/4 

70.883 

56.116 

1/4 

471.44 

962.52 

3/4 

1.7671 

.22089 

7/8 

74.663 

60.663 

V2 

490.87 

1  022.7 

13/16 

2.0739 

.28084 

5. 

78.540 

65.450 

3/4 

510.71 

1085.3 

7/8 

2.4053 

.35077 

1/8 

82.516 

70.482 

13. 

530.93 

1150.3 

15/16 

2.7611 

.43  143 

V4 

86.591 

75.767 

V4 

551.55 

1218.0 

1 

3.1416 

.52360 

3/8 

90.763 

81  .308 

1/2 

572.55 

1288.3 

Vl6 

3.5466 

.62804 

1/2 

95.033 

87.113 

3/4 

593.95 

1361.2 

1/8 

3.9761 

.7455 

5/8 

99.401 

93.189 

14. 

615.75 

1436.8 

3/16 

4.4301 

.8768 

3/4 

103.87 

99.541 

1/4 

637.95 

1515.1 

1/4 

4.9088 

.0227 

7/8 

108.44 

106.18 

1/2 

660.52 

1596.3 

5/16 

5.4119 

.1839 

6. 

113.10 

113.10 

3/4 

683.49 

1680.3 

3/8 

5.9396 

.3611 

1/8 

117.87 

120.31 

15. 

706.85 

1767.2 

7/16 

6.4919 

.5553 

1/4 

122.72 

127.83 

1/4 

730.63 

1857.0 

1/2 

7.0686 

.7671 

3/8 

127.68 

135.66 

1/2 

754.77 

1949.8 

9/16 

7.6699 

.9974 

1/2 

132.73 

143.79 

3/4 

779.32 

2045.7 

5/8 

8.2957 

2.2468 

5/8 

137.89 

152.25 

16. 

804.25 

2144.7 

,     U/16 

8.9461 

2.5161 

3/4 

143.14 

161.03 

1/4 

829.57 

2246.8 

I             Q/ 

3/4 

9.6211 

2.8062 

7/8 

148.49 

170.14 

1/2 

855.29 

2352.1 

13/16 

0.321 

3.1177 

7. 

153.94 

179.59 

3/4 

881.42 

2460.6 

'7/8 

1.044 

3.4514 

1/8 

159.49 

189.39 

17. 

907.93 

2572.4 

15/16 

1.793 

3.8083 

1/4 

165.13 

199.53 

1/4 

934.83 

2687.6 

2. 

2.566 

4.1888 

3/8 

1  70.87 

210.03 

1/2 

962.12 

2806.2 

1/16 

3.364 

4.5939 

1/2 

176.71 

220.89 

3/4 

989.80 

2928.2 

1/8 

4.186 

5.0243 

5/8 

182.66 

232.13 

18. 

1017.9 

3053.6 

3/16 

5.033 

5.4809 

3/4 

188.69 

243.73 

1/4 

1046.4 

3182.6 

1/4 

5.904 

5.9641 

7/8 

194.83 

255.72 

1/2 

1075.2 

3315.3 

5/16 

6.800 

6.4751 

8. 

201.06 

268.08 

3/4 

1  104.5 

3451.5 

3/8 

7.721 

7.0144 

V8 

207.39 

280.85 

19. 

1  134.1 

3591.4 

Vl6 

8.666 

7.5829 

1/4 

213.82 

294.01 

1/4 

1164.2 

3735.0 

1/2 

9.635 

8.1813 

3/8 

220.36 

307.58 

1/2 

1  194.6 

3882.5 

9/16 

0.629 

8.8103 

1/2 

226.98 

321.56 

3/4 

1225.4 

4033.7 

5/8 

1.648 

9.4708 

5/8 

233.71 

335.95 

20. 

1256.7 

4188.8 

U/16 

2.691 

0.154 

3/4 

240.53 

350.77 

V4 

1288.3 

4347.8 

3/4 

3.758 

0.889 

7/8 

247.45 

366.02 

1/2 

1320.3 

4510.9 

13/16 

4.850 

1.649 

9. 

254.47 

381.70 

3/4 

352.7 

4677.9 

7/8 

5.967 

2.443 

1/8 

261.59 

397.83 

21. 

385.5 

4849.1 

15/16 

7.109 

3.272 

1/4 

268.81 

414.41 

1/4 

418.6 

5024.3 

3. 

8.274 

4.137 

3/8 

270.12 

431.44 

1/2 

452.2 

5203.7 

1/16 

9.465 

5.039 

1/2 

283.53 

448.92 

3/4 

486.2 

5387.4 

1/8 

0.680 

5.979 

5/8 

291.04 

466.87 

23. 

520.5 

5575.3 

3/16 

1.919 

6.957 

3/4 

289.65 

485.31 

1/4 

555.3 

5767.6 

SPHERES. 


SPHERES  —  Continued. 


127 


Diam 

Sur- 
face. 

Vol- 
urne 

Diam 

Sur- 
face 

Vol- 
ume 

Diam 

Sur- 
face. 

Vol. 
ume. 

22  1/2 

1590.4 

5964. 

40  1/2 

5153.1 

34783 

70  1/2 

15615 

183471 

3/4 

1626.0 

6165.2 

41. 

5281.1 

36087 

71. 

15837 

187402 

23. 

1661.9 

6370.6 

V2 

5410.7 

37423 

1/2 

16061 

191389 

1/4 

1698.2 

6580.6 

43. 

5541.9 

38792 

73. 

16286 

195433 

1/2 

1735.0 

6795.2 

1/2 

5674.5 

40194 

V2 

16513 

199532 

3/4 

1772. 

7014.3 

43. 

5808.8 

41630 

73. 

16742 

203689 

24. 

1809.6 

7238.2 

1/2 

5944.7 

43099 

1/2 

16972 

207903 

V4 

1847.5 

7466.7 

44. 

6082.1 

44602 

74. 

17204 

212175 

1/2 

1885.8 

7700. 

1/2 

6221.2 

46141 

1/2 

17437 

216505 

3/4 

1924.4 

7938.3 

45. 

6361.7 

47713 

75. 

17672 

220894 

25. 

1963.5 

8181.3 

V2 

6503.9 

49321 

1/2 

17908 

225341 

V4 

2002.9 

8429.2 

46. 

6647.6 

50965 

76. 

18146 

229848 

!/2 

2042.6 

8682.0 

1/2 

6792.9 

52645 

V2 

18386 

234414 

3/4 

2083.0 

8939.9 

47. 

6939.9 

54362 

77. 

18626 

239041 

26. 

2123.7 

9202.8 

1/2 

7088.3 

56115 

V2 

18869 

243728 

1/4 

2164.7 

9470.8 

48. 

7238.3 

57906 

78. 

19114 

248475 

1/2 

2206.2 

9744.0 

1/2 

7389.9 

59734 

1/2 

19360 

253284 

3/4 

2248.0 

10022 

49. 

7543.1 

61601 

79. 

19607 

258155 

27. 

2290.2 

10306 

1/2 

7697.7 

63506 

1/2 

19856 

263088 

1/4 

2332.8 

10595 

50. 

7854.0 

65450 

80. 

20106 

268083 

1/2 

2375.8 

10889 

1/2 

8011.8 

67433 

1/2 

20358 

273147 

3/4 

2419.2 

11189 

51. 

8171.2 

69456 

81. 

20612 

278263 

28. 

2463.0 

11494 

1/2 

8332.3 

71519 

V2 

20867 

283447 

1/4 

2507.2 

11805 

52. 

8494.8 

73622 

83. 

2i<24 

288696 

1/2 

2551.8 

12121 

1/2 

8658.9 

75767 

1/2 

21382 

294010 

3/4 

2596.7 

12443 

53. 

8824.8 

77952 

83. 

21642 

299388 

29. 

2642.1 

12770 

1/2 

8992.0 

80178 

1/2 

21904 

304831 

1/4 

2687.8 

13103 

54. 

9160.8 

82448 

84. 

22167 

310340 

1/2 

2734.0 

13442 

V2 

9331.2 

84760 

1/2 

22432 

315915 

3/4 

2780.5 

13787 

55. 

9503.2 

87114 

85. 

22698 

321556 

30. 

2827.4 

14137 

V2 

9676.8 

89511 

1/2 

22966 

327264 

1/4 

2874.8 

14494 

56. 

9852.0  ' 

91953 

86. 

23235 

333039 

1/2 

2922.5 

14856 

V2 

10029 

94438 

1/2 

23506 

338882 

3/4 

2970.6 

15224 

57. 

10207 

96967 

87. 

23779 

344792 

31. 

3019.1 

15599 

1/2 

10387 

99541 

1/2 

24053 

350771 

1/4 

3068.0 

15979 

58. 

10568 

102161 

88. 

24328 

356819 

1/2 

3117.3 

16366 

1/2 

10751 

104826 

1/2 

24606 

362935 

3/4 

3166.9 

16758 

59. 

10936 

107536 

89. 

24885 

369122 

33. 

3217.0 

17157 

1/2 

11122 

110294 

1/2 

25165 

375378 

V4 

3267.4 

17563 

60. 

11310 

113098 

90. 

25447 

381704 

1/2 

3318.3 

17974 

V2 

11499 

115949 

1/2 

25730 

388102 

3/4 

3369.6 

18392 

61. 

11690 

118847 

91. 

26016 

394570 

33. 

3421.2 

18817 

1/2 

11882 

121794 

V2 

26302 

401109 

V4 

3473.3 

19248 

63. 

12076 

124789 

93. 

26590 

407721 

V2 

3525.7 

19685 

1/2 

12272 

127832 

1/2 

26880 

4  1  4405 

8/4 

3578.5 

20129 

63. 

12469 

130925 

93. 

27172 

421161 

34. 

3631.7 

20580 

1/2 

12668 

134067 

1/2 

27464 

427991 

1/4 

3685.3 

21037 

64. 

12868 

137259 

94. 

27759 

434894 

1/2 

3730.3 

21501 

1/2 

13070 

140501 

1/2 

28055 

441871 

35. 

3848.5 

22449 

65. 

13273 

143794 

95. 

28353 

448920 

V2 

3959.2 

23425 

1/2 

13478 

147138 

V2 

28652 

456047 

36. 

4071.5 

24429 

66. 

13685 

1  50533 

96. 

28953 

463248 

*/2 

4185.5 

25461 

1/2 

13893 

153980 

1/2 

29255 

470524 

37. 

4300.9 

26522 

67. 

14103 

157480 

97. 

29559 

477874 

V2 

4417.9 

27612 

V2 

14314 

161032 

1/2 

29865 

485302 

38. 

4536.5 

28731 

68. 

14527 

164637 

98. 

30172 

492808 

V2 

4656.7 

29880 

1/2 

14741 

168295 

1/2 

30481 

500388 

39. 

4778.4 

31059 

69. 

14957 

1  72007 

99. 

30791 

508047 

V2 

4901.7 

32270 

1/2 

15175 

175774 

V2 

31103 

515785 

40. 

5026.5 

33510 

70. 

15394 

1  79595 

00. 

31416 

523598 

128 


MATHEMATICAL   TABLES. 


NUMBER  OF  SQUARE  FEET  IN  PLATES  3  TO  32  FEET 
LONG,  AND   1  INCH  WIDE. 

For  other  widths, multiply  by  the  width  in  inches.  1  sq .  in.  =  0.00694/9  sq.  ft, 


Ft.  and 
Ins. 
Long. 

Ins. 
Long. 

Square 
Feet. 

Ft.  and 
Ins. 
Long; 

Ins. 
Long. 

Square 
Feet. 

Ft.  and 
Ins. 
Long. 

Ins. 
Long. 

Square 
Feet. 

3.  0 

36 

.25 

7.  10 

94 

.6528 

12.  8 

152 

.056 

37 

.2569 

11 

95 

.6597 

9 

153 

.063 

2 

38 

.2639 

8.  0 

96 

.6667 

10 

154 

.069 

3 

39 

.2708 

1 

97 

.6736 

11 

155 

,076 

4 

40 

.2778 

2 

98 

.6806 

13.  0 

156 

,083 

5 

41 

.2847 

3 

99 

.6875 

1 

157 

09 

6 

42 

.2917 

4 

100 

.6944 

2 

158 

.097 

7 

43 

.2986 

5 

101 

.7014 

3 

159 

.104 

8 

44 

.3056 

6 

102 

.7083 

4 

160 

.1  14 

9 

45 

.3125 

7 

103 

.7153 

5 

161 

.ua 

10 

46 

.3194 

8 

104 

.7222 

6 

162 

.125 

11 
4.  0 

47 
48 

.3264 
.3333 

9 
10 

105 
106 

.7292 
.7361 

7 
8 

163 
•164 

.13.? 
.  1  3V 

49 

.3403 

11 

107 

.7431 

9 

165 

.146 

2 

50 

.3472 

9.  0 

108 

.75 

10 

166 

.153 

3 

51 

.3542 

1 

109 

.7569 

11 

167 

.159 

4 

52 

.3611 

2 

110 

.7639 

14.  0 

168 

.167 

5 

53 

.3681 

3 

111 

.7708 

1 

169 

.174 

6 

54 

.375 

4 

112 

.7778 

2 

170 

.181 

7 

55 

.3819 

5 

113 

.7847 

3 

171 

.188 

8 

56 

.3889 

6 

114 

.7917 

4 

172 

.194 

9 

57 

.3958 

7 

115 

.7986 

5 

173 

.201 

10 

58 

.4028 

8 

116 

.8056 

6 

174 

.208 

It 

59 

.4097 

9 

117 

.8125 

7 

175 

.215 

5.  0 

60 

.4167 

10 

•  118 

.8194 

8 

176 

.222 

61 

.4236 

11 

119 

.8264 

9 

177 

.229 

2 

62 

.4306 

10.  0 

120 

.8333 

10 

178 

.236 

3 

63 

.4375 

121 

.8403 

11 

179 

.243 

4 

64 

.4444 

2 

122 

.8472 

15.  0 

180 

.25 

5 

65 

.4514 

3 

123 

.8542 

181 

.257 

6 

66 

.4583 

4 

124 

.8611 

2 

182 

.264 

7 

67 

.4653 

5 

125 

.8681 

3 

183 

.271 

8 

68 

.4722 

6 

126 

.875 

4 

184 

.278 

9 

69 

.4792 

7 

127 

.8819 

5 

185 

.285 

10 

70 

.4861 

8 

128 

.8889 

6 

186 

.292 

11 

71 

.4931 

9 

129 

.8958 

7 

187 

.299 

8.  0 

72 

.5 

10 

130 

.9028 

8 

188 

.306 

1 

73 

.5069 

11 

131 

.9097 

9 

189 

.313 

2 

74 

.5139 

11.  0 

132 

.9167 

10 

190 

.319 

3 

75 

.5208 

133 

.9236 

11 

191 

.326 

4 

76 

.5278 

2 

134 

.9306 

16.  0 

192 

.333 

5 

77 

.5347 

3 

135 

.9375 

1 

193 

.34 

6 

78 

.5417 

4 

136 

.9444 

2 

194 

.347 

7 

79 

.5486 

5 

137 

.9514 

3 

195 

.354 

8 

80 

.5556 

6 

138 

.9583 

4 

196 

.361 

9 

81 

.5625 

7 

139 

.9653 

5 

197 

.368 

10 

82 

.5694 

8 

140 

.9722 

6 

198 

.375 

11 

83 

.5764 

9 

141 

.9792 

7 

199 

.382 

7.  0 

84 

.5834 

10 

142 

.9861 

8 

200 

.389 

85 

.5903 

11 

143 

.9931 

9 

201 

.396 

2 

86 

.5972 

12.  0 

144 

.000 

10 

202 

.403 

3 

87 

.6042 

145 

.007 

11 

203 

.41 

4 

88 

.6111 

2 

146 

.014 

17.  0 

204 

.417 

5 

89 

.6181 

3 

147 

.021 

1 

205 

.424 

6 

90 

.625 

4 

148 

.028 

2 

206 

.431 

7 

91 

.6319 

5 

149 

.035 

3 

207 

.438 

8 

92 

.6389 

6 

150 

1.042 

4 

208 

.444 

9 

93 

.6458 

7 

151 

1.049 

5 

209 

K451 

NUMBER   OF   SQUARE    FEET   IN   PLATES. 


120 


SQUARE  FEET  IN  PLATES.  —  Continued. 


Ft.  and 
Ins. 
Long. 

Ins. 
Long. 

Square 
Feet. 

Ft.  and 
Ins. 
Long. 

Ins. 
Long. 

Square 
Feet. 

Ft.  and 
Ins.. 
Long. 

Ins. 
Long 

Square 
Feet. 

17.  6 

210 

1.458 

22.  5 

269 

1.868 

27.  4 

328 

2.278 

7 

211 

1.465 

6 

270 

1.875 

5 

329 

2.285 

8 

212 

1.472 

7 

271 

1.882 

6 

330 

2.292 

9 

213 

1.479 

8 

272 

1.889 

7 

331 

2.299 

10 

214 

1.486 

9 

273 

1.896 

8 

332 

2.306 

11 

215 

1.493 

10 

274 

1.903 

9 

333 

2.313 

18.  0 

216. 

1.5 

11 

275 

1.91 

10 

334 

2.319 

217 

1.507 

23.  0 

276 

1.917 

11 

335 

2.326 

2 

218 

1.514 

277 

1.924 

28.  0 

336 

2.333 

3 

219 

1.521 

2 

278. 

1.931 

1 

337 

2.34 

4 

220 

1.528 

3 

279 

1.938 

2 

338 

2.347 

5 

221 

1.535 

4 

280 

1.944 

3 

339 

2.354 

6 

222 

1.542 

5 

281 

1.951 

4 

340 

2.361 

7 

223 

1.549 

6 

282 

1.958 

5 

341 

2.368 

8 

224 

1.556 

7 

283 

1.965 

6 

342 

2.375 

9 

225 

1.563 

8 

284 

1.972 

7 

343 

2.382 

10 

226 

1.569 

9 

285 

1.979 

8 

344 

2.389 

11 

227 

1.576 

10 

286 

1.986 

9 

345 

2.396 

19.  0 

228 

1.583 

11 

287 

1.993 

10 

346 

2.403 

229 

1.59 

24.  0 

288 

2. 

11 

347 

2.41 

2 

230 

1.597 

1 

289 

2.007 

29.  0 

348 

2.417 

3 

231 

1.604 

2 

290 

2.014 

349 

2.424 

4 

232 

1.611 

3 

291" 

2.021 

2 

350 

2.431 

5 

233 

1.618 

4 

292 

2.028 

3 

351 

2.438 

6 

234 

1.625 

5 

293 

2.035 

4 

352 

2.444 

7 

235 

1.632 

6 

294 

2'.042 

5 

353 

2.451 

8 

236 

1.639 

7 

295 

2.049 

6 

354 

2.458 

9 

237 

1.645 

8 

296 

2.056 

7 

355 

2.465 

10 

238 

1.653 

9 

297 

2.063 

8 

356 

2.472 

11 

239 

1  .659 

10 

298 

2.069 

9 

357 

2.479 

20.  0 

240 

1.667 

11 

299 

2.076 

10 

358 

2.486 

241 

1.674 

25.  0 

300 

*2.083 

11 

359 

2.493 

2 

242 

1.681 

1 

301 

2.09 

30.  0 

360 

2.5 

3 

243 

1.688 

2 

302 

2.097 

1 

361 

2.507 

4 

244 

1.694 

3 

303 

2.104 

2 

362 

2.514 

5 

245 

1.701 

4 

304 

2.111 

3 

363 

2.521 

6 

246 

1.708 

5 

305 

2.118 

4 

364 

2.528 

7 

247 

1.715 

6 

306 

2.125 

5 

365 

2.535 

8 

248 

1.722 

7 

307 

2.132 

6 

366 

2.542 

9 

249 

1.729 

8 

308 

2.139 

7 

367 

2.549 

10 

250 

1.736 

9 

309 

2.146 

8 

368 

2.556 

II 

251 

1.743 

10 

310 

2.153 

9 

369 

2.563 

21.  0 

252 

1.75 

11 

311 

2.16 

10 

370 

2.569 

253 

1.757 

26.  0 

312 

2.167 

11 

371 

2.576 

2 

254 

1.764 

313 

2.174 

31.  0 

372 

2.583 

3 

255 

1.771 

2 

314 

2.18V 

373 

2.59 

4 

256 

1.778 

3 

315 

2.188 

2 

374 

2.597 

5 

257 

1.785 

4 

316 

2.194 

3 

375 

2.604 

6 

258 

1.792 

5 

317 

2.201 

4 

376 

2.611 

7 

259 

1.799 

6 

318 

2.208 

5 

377 

2.618 

8 

260 

1.806 

7 

319 

2.215 

6 

378 

2.625 

9 

261 

1.813 

8 

320 

2.222 

7 

379 

2.632 

10 

262 

1.819 

9 

321 

2.229 

8 

380 

2.639 

11 

263 

1.826 

10 

322 

2.236 

9 

381 

2.646 

23.0 

264 

1.833 

11 

323 

2.243 

10 

382 

2.653 

1 

265 

1.84 

27.  0 

324 

2.25 

11 

383 

2.66 

2 

266 

1.847 

325 

2.257 

32.  0 

384 

2.667 

3 

267 

1.854 

2 

326 

2.264 

1 

385 

2.674 

4 

268 

1.861 

3 

327 

2.271 

2 

386 

2.681 

130 


MATHEMATICAL   TABLES. 


GALLONS  AND   CUBIC  FEET, 

United  States  Gallons  in  a  given  Number  of  Cubic  Feet. 

1  cubic  foot  =  7. 4805 19  U.S.  gallons;  1  gallon  =  231  cu.ir  .  =  0.13368056cu. ft. 


Cubic  Ft. 

Gallons. 

Cubic  Ft. 

Gallons. 

Cubic  Ft. 

Gallons. 

0.1 

0.75 

50 

374.0 

8,000 

59,844.2 

0.2 

1.50 

60 

448.8 

9,000 

67,324.7 

0.3 

2.24 

70 

523.6 

10,000 

74,805.2 

0.4 

2.99 

80 

598.4 

20,000 

.••      149,610.4 

0.5 

3.74 

90 

673.2 

30,000 

224,415.6 

0.6 

4.49 

100 

748.0 

40,000 

299,220.8 

0.7 

5.24 

200 

1,496.1 

50,000 

374,025.9 

0.8 

5.98 

300 

2,244.2 

60,000 

448,831.1 

0.9 

6.73 

400 

2,992.2 

70,000 

523,636.3 

1 

7.48 

500 

3,740.3 

80,000 

598,441.5 

2 

14.96 

600 

4,488.3 

90,000 

673,246. 

3 

22.44 

700 

5,236.4 

100,000 

748,051.9 

4 

29.92 

800 

5,984.4 

200,000 

1,496,103.8 

5 

37.40 

900 

6,732.5 

300,000 

2,244,155.7 

6 

44.88 

1,000 

7,480.5 

400,000 

2,992,207.6 

7 

52.36 

2,000 

14,961.0 

500,000 

3,740,259.5 

8 

59.84 

3,000 

22,441.6 

600,000 

4,488,311.4 

9 

67.32 

4,000 

29,922.1 

700,000 

5,236,363.3 

10 

74.80 

5,000 

37,402.6 

800  000 

5,984,415.2 

20 

149.6 

6,000 

44,883.1 

900,000 

6,732,467.1 

30 

224.4 

7,000 

52,363.6 

1,000,000 

7,480,519.0 

40 

299.2 

Cubic  Feet  in  a  given  Number  of  Gallons. 


Gallons. 

Cubic  Ft. 

Gallons. 

Cubic  Ft. 

Gallons. 

Cubic  Ft. 

1 
2 

.134 
.267 

1,000 
2,000 

133.681 
267.361 

1,000,000 
2,000,000 

133,680.6 
267,361.1 

3 

.401 

3,000 

401.042 

3,000,000 

401,041.7 

4 

.535 

4,000 

534.722 

4,000,000 

534,722.2 

5 

.668 

5,000 

668.403 

5,000,000 

668,402.8 

6 

.802 

6,000 

802.083 

6,000,000 

802,083.3 

7 

.936 

7,000 

935.764 

7,000,000 

935,763.9 

8 

1.069 

'    8,000 

1,069.444 

8,000,000 

1,069,444.4 

9 

1.203 

9,000 

1,203.125 

9,000,000 

1,203,125.0 

10 

1.337 

10,000 

1,336.806 

10,000,000 

1,336,805.6 

Cubic  Feet  per  Second,  Gallons  in  24  hours,  etc. 

1/60  I  1.5472  2.2800 

1  60  92.834  133.681 

7.480519  448.83  694.444  1,000. 

10,771.95  646,317  1,000,000  1,440,000 

62.355      3741.3          5788.66  8335.65 


Cu.  ft.  per  sec. 
Cu.  ft.  per  min. 
U.  S*  Gals,  per  min. 

"  "  "  24  hrs. 
Pounds  of  water  ) 
(at  62°  F.)  per  min.  J 


The  gallon 'is  a  troublesome  and  unnecessary  measure.  If  hydraulic 
engineers  and  pump  manufacturers  would  stop  using  it,  and  use  cubig 
Jeet  instead,  many  tedious  calculations  would  be  saved. 


CAPACITY   OF   CYLINDKICAL   VESSELS. 


131 


CONTENTS  IN   CUBIC  FEET  AND  U.  S.  GALLONS  OF  PIPES 

AND   CYLINDERS  OF  VARIOUS  DIAMETERS  AND  ONE 

FOOT  IN  LENGTH. 

1  gallon  =  231  cubic  inches.     1  cubic  foot  =  7.4805  gallons. 


For  1  Foot  in 

For  1  Foot  in 

For  1  Foot  in 

d 

Length. 

.S 

Length. 

d 

Length. 

5  as 

0>  3? 

fc« 

•»•»  2 
§•§ 

Cu.Ft. 

U.S. 

-t~    O 

oj^j 
Jo 

Cu.Ft. 

U.S. 

•S  2 

d  « 

Cu.Ft. 

U.S. 

c  c 

also 

Gals., 

d 

also 

Gals., 

d  d 

also 

Gals.. 

Q 

Area  in 

231 

p 

Area  in 

231 

Area  in 

231 

M 

Sq.Ft. 

Cu.In. 

Sq.Ft. 

Cu.In. 

. 

Sq.Ft. 

Cu.  In. 

V4 

.0003 

.0025 

63/4 

.2485 

1.859 

19 

1.969 

14.73 

5/16 

.0005 

.004 

7 

.2673 

1.999 

191/2 

2074 

15.51 

3/8 

.0008 

.0057 

7V4 

.2867 

2.145 

20 

2.182 

16.32 

7/16 

.001 

.0078 

71/2 

.3068 

2.295 

201/2 

2.292 

17.15 

1/2 

.0014 

.0102 

73/4 

.3276 

2.45 

21 

2.405 

17.99 

9/16 

.0017 

.0129 

8 

.3491 

2.611 

2U/2 

2.521 

18.86 

5/8 

.0021 

.0159 

8l/4 

.3712 

2.777 

22 

2.640 

19.75 

11/16 

.0026 

;0193 

81/2 

.3941 

2.948 

221/2 

2.761 

20.66 

3/4 

.0031 

.0230 

83/4 

.4176 

3.125 

23 

2.885 

21.58 

«/16 

.0036 

.0269 

9 

.4418 

3.305 

231/2 

3.012 

22.53 

7/8 

.0042 

.0312 

91/4 

.4667 

3.491 

24 

3.142 

23.50 

15/16 

.0048 

.0359 

91/2 

.4922 

3.682 

25 

3.409 

25.50 

1 

.0055 

.0408 

93/4 

.5185 

3.879 

26 

3.687 

27.58 

U/4 

.0085 

.0638 

10 

.5454 

4.08 

27 

3.976 

29.74 

U/2 

.0123 

.0918 

101/4 

.5730 

4.286 

28 

4.276 

31.99 

13/4 

.0167 

.1249 

101/2 

.6013 

4.498 

29 

4.587 

34.31 

2 

.0218 

.1632 

103/4 

.6303 

4.715 

30 

4.909 

36.72 

2V4    - 

.0276 

.2066 

11 

.66 

4.937 

31 

5.241 

39.21 

21/2 

.0341 

.2550 

111/4 

.6903 

5.164 

32 

5.585 

41.78 

23/4 

.0412 

.3085 

111/2 

.7213 

5.396 

33 

5.940 

44.43 

3 

.0491 

.3672 

113/4 

.7530 

5.633 

34 

6.305 

47.16 

31/4 

.0576 

.4309 

12 

.7854 

5.875 

35 

6.681 

49.98 

31/2 

.0668 

.4998 

121/2 

.8522 

6.375 

36 

7.069 

52.88 

33/4 

.0767 

.5738 

13 

.9218 

6.895 

37 

7.467 

55.86 

4 

.0873 

.6528 

13V2 

.994 

7.436 

38 

7.876 

58.92 

41/4 

.0985 

.7369 

14 

1.069 

7.997 

39 

8.296 

62.06 

4V2 

.1104 

.8263 

141/2 

1.147 

8.578 

40 

8.727 

65.28 

43/4 

.1231 

.9206 

15 

1.227 

9.180 

41 

9.168 

68.58 

5 

.1364 

.020 

15l/2 

1.310 

9.801 

42 

9.621 

71.97 

5V4 

.1503 

.125 

16 

1.396 

10.44 

43 

10.085 

75.44 

5i/2 

.1650 

.234 

161/2 

.485 

11.11 

44 

10.559 

78.99 

53/4 

.1803 

.349 

17 

.576 

11.79 

45 

11.045 

82.62 

6 

.1963 

.469 

171/2 

.670 

12.49 

46 

11.541 

86.33 

61/4 

.2131 

.594 

18 

.768 

13.22 

47 

12.048 

90.10 

61/2 

.2304 

.724 

18l/2 

.867 

13.96 

48 

12.566 

94.00 

To^find  the  capacity  of  pipes  greater  than  the  largest  given  in  the  table, 


aer  n  any  o  z, 

in  cubic  feet  by  621/4  or  the  gallons  by  8 1/3,  or,  if  a  closer  approximation  is 
required,  by  the  weight  of  a  cubic  foot  of  water  at  the  actual  temperature 
in  the  pipe. 

Given  the  dimensions  of  a  cylinder  in  inches,  to  find  its  capacity  in  U.  8. 
gallons:  Square  the  diameter,  multiply  by  the  length  and  by  0.0034.  If  d= 

diameter,  I  -  length,  gallons-  d*  X  °^54  X  *  - 0.0034  &  1.  If  D  and  L  are 
in  feet,  gallons  -  5.875  D*L. 


.132 


MATHEMATICAL  TABLES. 


CYLINDRICAL,  VESSELS,  TANKS,  CISTERNS,  ETC. 

Diameter  In  Feet  and  Inches,  Area  in  Square  Feet,  and  U.  S, 
Gallons  Capacity  for  One  Foot  in  Depth. 


1  gallon  =  231  cubic  inches  = 


1  cubic  foot 

7.4805 


'  0.13368  cubic  feet. 


Diam. 

Area. 

Gals. 

Diam. 

Area. 

Gals. 

Diam. 

Area. 

Gals. 

Ft.  In. 

Sq.ft. 

1  foot 
depth. 

Ft.  In. 

Sq.  ft. 

1  foot 
depth. 

Ft.  In. 

Sq.ft. 

1  foot 
depth. 

1 

.785 

5.87 

5    8 

25.22 

188  .66 

19 

283  .53 

2120.9 

1 

.922 

6.89 

5    9 

25.97 

194.25 

19    3 

291.04 

2177.1 

2 

.069 

8.00 

510 

26.73 

199.92 

19    6 

298.65 

2234.0 

3 

.227 

9.18 

5  11 

27.49 

205.67 

19    9 

306.35 

2291.7 

A 

.396 

10.44 

6 

28.27 

211.51 

20 

314.16 

2350.1 

5 

.576 

11.79 

6    3 

30.68 

229.50 

20    3 

322.06 

2409.2 

6 

.767 

13.22 

6    6 

33.18 

248.23 

20    6 

330.C6 

2469.1 

7 

.969 

14.73 

6    9 

35.78 

267.69 

20    9 

338.16 

2529,6 

8 

2.182 

16.32 

7 

38.48 

287.88 

21 

346.36 

2591.0 

9 

2.405 

17.99 

7    3 

41.28 

308.81 

21    3 

354.66 

2653.0 

10 

2.640 

19.75 

7    6 

44.18 

330.48 

21    6 

363.05 

2715.8 

11 

2.885 

21.58 

7    9 

47.17 

352.88 

21    9 

371.54 

2779.3 

3.142 

23.50 

8 

50.27 

376.01 

22 

380.13 

2843.6 

1 

3.409 

25.50 

8    3 

53.46 

399.88 

22    3 

388.82 

2908.6 

2      2 

3.687 

27.58 

8    6 

56.75 

424.48 

22    6 

397.61 

2974.3 

2      3 

3.976 

29.74 

8    9 

60.13 

449.82 

22    9 

406.49 

3040.8 

2      4 

4.276 

31.99 

9 

63.62 

475.89 

23 

415.48 

3108.0 

2      5 

4.587 

34.31 

9    3 

67.20 

502.70 

23    3 

424.56 

3175.9 

2      6 

4.909 

36.72 

9    6 

70.88 

530.24 

23    6 

433.74 

3244.6 

2      7 

5.241 

39.21 

9    9 

74.66 

558.51 

23    9 

443.01 

3314.0 

2      8 

5.585 

41.78 

10 

78.54 

587.52 

24 

452.39 

3384.1 

2      9 

5.940 

44.43 

10    3 

82.52 

617.26 

24    3 

461.86 

3455.0 

2     10 

6.305 

47.16 

10-   6 

86.59 

647.74 

24    6 

471.44 

3526.6 

2    11 

6.681 

49.98 

10    9 

90.76 

678.95 

24    9 

481.11 

3598.9 

3 

7.069 

52.88 

11 

95.03 

710.90 

25 

490.87 

3672.0 

1 

7.467 

55.86 

11    3 

99.40 

743.58 

25    3 

500.74 

3745.8 

2 

7.876 

58.92 

11    6 

103.87 

776.99 

25    6 

510.71 

3820.3 

3 

8.296 

62.06 

11    9 

108.43 

811.14 

25    9 

520.77 

3895.6 

A 

8.727 

65.28 

12 

113.10 

846.03 

26 

530.93 

3971.6 

5 

9.168 

68.58 

12    3 

117.86 

881.65 

26    3 

541.19 

4048.4 

6 

9.621 

71.97 

12    6 

122.72 

918.00 

26    6 

551.55 

4125.9 

7 

10.085 

75.44 

12    9 

127.68 

955.09 

26    9 

562.00 

4204.  1 

8 

10.559 

78.99 

13 

132.73 

992.91 

27 

572.56 

4283.0 

9 

1  1  .045 

82.62 

13    3 

137.89 

1031.5 

27    3 

583.21 

4362.7 

10 

11.541 

86.33 

13    6 

143.14 

1070.8 

27    6 

593.96 

4443.1 

11 

12.048 

90.13 

13    9 

1  48.49 

1110.8 

27    9 

604.81 

4524.3 

12.566 

94.00 

14 

153.94 

1151.5 

28 

615.75 

4606.2 

1 

13.095 

97.96 

14    3 

159.48 

1193.0 

28    3 

626.80 

4688.8 

2 

13.635 

102.00 

14    6 

165.13 

1235.3 

28    6 

637.94 

4772.1 

3 

14.186 

106.12 

14    9 

170.87 

1278.2 

28    9 

649.18 

4856.2 

4 

14.748 

110.32 

15 

176.71 

1321.9 

29 

660.52 

4941.0 

5 

15.321 

114.61 

15    3 

182.65 

1366.4 

29   3 

67  1  .96 

5026.6 

6 

15.90 

118.97 

15    6 

188.69 

1411.5 

29    6 

683.49 

5112.9 

7 

16.50 

123.42 

15    9 

194.83 

1457.4 

29    9 

695.13 

5199.9 

8 

17.10 

127.95 

16 

201.06 

1504.1 

30 

706.86 

5287.7 

9 

17.72 

132.56 

46    3 

207.39 

1551.4 

30    3 

718.69 

5376.2 

10 

18.35 

137.25 

16    6 

213.82 

1  599.5 

30    6 

730.62 

5465.4 

11 

18.99 

142.02 

16    9 

220.35 

1648.4 

30    9 

742.64 

5555.4 

19.63 

146.88 

17 

226.98 

1697.9 

31 

754.77 

5646.1 

1 

20.29 

151.82 

17    3 

233.71 

1748.2 

31    3 

766.99 

5737.5 

2 

20.97 

156.83 

17    6 

240.53 

1799.3 

31    6 

779.31 

5829.7 

3 

21.65 

161.93 

17    9 

247.45 

1851.1 

31    9 

791.73 

5922.6 

4 

22.34 

167.12 

18 

254.47 

1903.6 

32 

804.25 

6016.2 

5 

23.04 

172.38 

18    3 

261.59 

1956.8 

32    3 

816.86 

6110.6 

6 

23.76 

177.  ,72 

18    6 

268.80 

2010.8 

32    6 

829.58 

6205.7 

7 

24.48 

183.15 

18   9 

276.12 

2065.5 

32    9 

842.39 

6301.5 

CAPACITIES   OF  RECTANGULAR  TANKS. 


133 


CAPACITIES  OF  RECTANGULAR  TANKS  IN  U.  S. 
GALLONS,  FOB  EACH  FOOT  IN  DEPTH. 

1  cubic  foot  =-  7.4805  U.  S.  gallons 


Vidth 
of 
Fank. 

Length  of  Tank. 

feet. 

2 

ft.  in. 
2  6 

feet. 
3 

ft.  in. 
3  6 

feet. 
4 

ft.  in. 
4     6 

feet. 
5 

ft.  in. 
5  6 

feet. 
6 

ft.  in. 
6  6 

feet. 
7 

t.   in. 
2    6 

3    6 
4 

4    6 

5    6 
6 
6    6 

7 

29.92 

37.40 
46.75 

44.88 
56.10 
67.32 

52.36 
65.45 
78.54 
91.64 

59.84 
74.80 
89.77 
104.73 
119.69 

67.32 
84.16 
1  00.99 
117.82 
134.65 

151.48 

74.81 
93.51 
112.21 
130.91 
149.61 

168.31 
187.01 

82.29 
102.86 
123.43 
144.00 
164.57 

185.14 
205.71 
226.28 

89.77 
112.21 
134.65 
157.09 
179.53 

201.97 
224.41 
246.86 
269.30 

97.25 
121.56 
145.87 
170.18 
194.49 

218.80 
243.11 
267.43 
291.74 
316.05 

104.73 
130.91 
157.09 
183.27 
209.45 

235.62 
261.82 
288.00 
314.18 
340.36 

366.54 

... 

Width 
of 
Tank. 

Length  of  Tank. 

ft.  in 
7   6 

feet. 
8 

ft.  in. 
8  6 

feet. 
9 

ft.  in 
9  6 

feet. 
10 

ft.  in 
10  6 

feet. 
11 

ft.  in 
11  6 

feet. 
13 

179.53 
224.41 
269.30 
314.18 
359.06 

403.94 
448.83 
493.  7  1 
538.59 
583.47 

628.36 
673.24 
718.12 
763.00 
807.89 

852.77 
897.66 
942.56 
987.43 
1032.3 

1077.2 

ft.  in 
2 
2  6 
3 
3  6 
4 

4  6 

5  6 
6 
6  6 

7  6 
8 
8  6 
9 

9  6 
10 
10  6 
11 
11  6 

12 

112.21 
140.26 
168.31 
196.36 
224.41 

25247 
280.52 
308.57 
336.62 
364.67 

392.72 
420.78 

119.69 
149.61 
179.53 
209.45 
239.37 

269.30 
299.22 
329.14 
359.06 
388.98 

418.91 
448.83 
478.75 

127.17 
158.96 
190.75 
222.54 
254.34 

286.13 
317.92 
349.71 
381.50 
413.30 

445.09 
476.88 
508.67 
540.46 

134.65 
168.31 
202.97 
235.63 
269.30 

302.96 
336.62 
370.28 
403.94 
437.60 

471.27 
504.93 
538.59 
572.25 
605.92 

142.13 
177.66 
213.19 
248.73 
284.26 

319.79 
355.32 
390.85 
426.39 
461.92 

497.45 
532.98 
568.51 
604.05 
639.58 

675.11 

149.61 
187.01 
224.41 
261.82 
299.22 

336.62 
374.03 
411.43 
448.83 
486.23 

523.64 
561.04 
598.44 
635.84 
673.25 

710.65 
748.05 

157.09 
196.36 
235.63 
274.90 
314.18 

353.45 
392.72 
432.00 
471.27 
510.54 

549.81 
589.08 
628.36 
667.63 
706.90 

746.17 

785.45 
824.73 

164.57 
205.71 
246.86 
288.00 
329.14 

370.28 
411.43 
452.57 
493.71 
534.85 

575.99 
617.14 
658.28 
699.42 
740.56 

781.71 
822.86 
864.00 
905.14 

172.05 
215.06 
258.07 
301.09 
344.10 

387.11 
430.13 
473.14 
516.15 
559.16 

602.18 
645.19 
688.20 
731.21 
774.23 

817.24 
860.26 
903.26 
946.27 
989.29 

134 


MATHEMATICAL   TABLES. 


NUMBER  OF  BARRELS    (31  1-3  GALLONS)  IN 
CISTERNS  AND  TANKS. 


I  barrel  =  31^  gallons  > 


31.5X  231 
1728 


=  4.21094  cu.  ft.  Reciprocal -0.2  37 477 


Diameter  in  Feet. 


Feet. 

5 

6 

7 

8 

9 

10 

11 

13 

13 

14 

, 

4.663 

6.714 

9.139 

11.937 

15.108 

18.652  , 

>2.569 

26.859 

31.522 

36.557 

5 

23.3 

33.6 

45.7 

59.7 

75.5 

93.3 

12.8 

134.3 

157.6 

182.8 

6 

28.0 

40.3 

54.8 

71.6 

90.6 

111.9 

35.4 

161.2 

189.1 

219.3 

7 

32.6 

47.0 

64.0 

83.6 

105.8 

130.6 

58.0 

188.0 

220.7 

255.9 

8 

37.3 

53.7 

73.1 

95.5 

120.9 

149.2 

80.6 

214.9 

252.2 

292.5 

9 

42.0 

60.4 

82.3 

107.4 

136.0 

167.9    ; 

>03.1 

241.7 

283.7 

329.0 

10 

46.6 

67.1 

91.4 

119.4 

151.1 

186.5    ; 

Z25.7 

268.6 

315.2 

365.6 

11 

51.3 

73.9 

100.5 

131.3 

166.2 

205.2    ; 

548.3 

295.4 

346.7 

402.1 

12 

56.0 

80.6 

109.7 

143.2 

181.3 

223.8    ; 

570.8 

322.3 

378.3 

438.7 

13 

60.6 

87.3 

118.8 

155.2 

196.4 

242.5    : 

593.4 

349.2 

409.8 

475.2 

14 

65.3 

94.0 

127.9 

167.1 

211.5 

261.1    I 

16.0 

376.0 

44  K3 

511.8 

15 

69.9 

100.7 

137.1 

179.1 

226.6 

279.8    2 

38.5 

402.9 

472.8 

548.4 

16 

74.6 

107.4 

146.2 

191.0 

241.7 

298.4    2 

61.1 

429.7 

504.4 

584.9 

17 

79.3 

114.1 

155.4 

202.9 

256.8 

317.1     2 

83.7 

456.6 

535.9 

621.5 

18 

83.9 

120.9 

164.5 

214.9 

271.9 

335.7    ^ 

K)6.2 

483.5 

567.4 

658.0 

19 

88.6 

127.6 

173.6 

226.8 

287.1 

354.4    ^ 

128.8 

510.3 

598.9 

694.6 

20 

93.3 

134.3 

182.8 

238.7 

302.2 

373.0    * 

151.4 

537.2 

630.4 

731.1 

Depth 
in 

Diameter  in  Feet. 

Feet. 

15 

16 

17 

18 

19 

20 

21 

22 

1 

41.966 

47.748 

53.903 

60.431 

67.33. 

I      74.606 

82.253 

90.273 

5 

209.8 

238.7 

269.5 

302.2 

336.7 

373.0 

411.3 

451.4 

6 

251.8 

286.5 

323.4 

362.6 

404.0 

447.6 

493.5 

541.6 

7 

293.8 

334.2 

377.3 

423.0 

471.3 

522.2 

575.8 

631.9 

8 

335.7 

382.0 

431.2 

483.4 

538.7 

596.8 

658.0 

722.2 

9 

377.7 

429.7 

485.1 

543.9 

606.0 

671.5 

740.3 

812.5 

10 

419.7 

477.5 

539.0 

604.3 

673.3 

746.1 

822.5 

902.7 

11 

461.6 

525.2 

592.9 

664.7 

740.7 

820.7 

904.8 

993.0 

12 

503.6 

573.0 

646.8 

725.2 

808.0 

895.3 

987.0 

1083.3 

13 

545.6 

620.7 

700.7 

785.6 

875.3 

969.9 

1069.3 

1173.5 

14 

587.5 

668.5 

754.6 

846.0 

942.6 

1044.5 

1151.5 

1263.8 

15 

629.5 

716.2 

808.5 

906.5 

1010.0 

1119.1 

1233.8 

1354.1 

16 

671.5 

764.0 

862.4 

966.9 

1077.3 

1193.7 

1316.0 

1444.4 

17 

713.4 

811.7 

916.4 

1027.3 

1144.6 

1268.3 

1398.3 

1534.5 

18 

755.4 

859.5 

970.3 

1087.8 

1212.0 

1342.9 

1480.6 

1624.9 

19 

797.4 

907.2 

1024.2 

1148.2 

1279.3 

1417.5 

1562.8 

1715.2 

20 

839.3 

955.0 

1078.1 

1208.6 

1346.6 

1492.1 

1645.1 

1805.5 

i 

LOGARITHMS    OF   NUMBERS. 


135 


NUMBER  OF  BARBELS   (31  1-2  GALLONS)  IN  CISTERNS 
AND  TANKS.  —  Continued. 


Depth 
in 
Feet. 

Diameter  in  Feet. 

23 

24 

25 

26 

27 

28 

29 

30 

1 
5 

98.666 
493.3 

107.432 
537.2 

116.571 
582.9 

126.083 
630.4 

135.968 
679.8 

146.226 
731.1 

156.858 
784.3 

167.863 
839.3 

6 

592.0 

644.6 

699.4 

756.5 

815.8 

877.4 

941.1 

1007.2 

7 

690.7 

752.0 

316.0 

882.6 

951.8 

1023.6 

1098.0 

1175.0 

8 

789.3 

859.5 

932.6 

1008.7 

1087.7 

1169.8 

1254.9 

1342.9 

9 

888.0 

966.9 

1049.1 

1134.7 

1223.7 

1316.0 

1411.7 

1510.8 

to 

986.7 

1074.3 

1165.7 

1260.8 

1359.7 

1462.2 

1  568.6 

1678.6 

11 

1085.3 

1  181.8 

1282.3 

1386.9 

1495.6 

1608.5 

1725.4 

1846.5 

12 

1184.0 

1289.2 

1398.8 

1513.0 

1631.6 

1754.7 

1882.3 

2014.4 

13 

1282.7 

1396.6 

1515.4 

1639.1 

1767.6 

1900.9 

2039.2 

2182.2 

14 

1381.3 

1504.0 

1632.0 

1765.2 

1903.6 

2047.2 

2196.0 

2350.1 

15 

1480.0 

1611.5 

1  748.6 

1891.2 

2039.5 

2193.4 

2352.9 

2517.9 

16 

1578.7 

1718.9 

1865.1 

2017.3 

2175.5 

2339.6 

2509.7 

2685.8 

17 

1677.3 

1826.3 

1981.7 

2143.4 

2311.5 

2485.8 

2666.6 

2853.7 

18 

1776.0 

1933.8 

2098.3 

2269.5 

2447.4 

2632.0 

2823.4 

3021.5 

19 

1874.7 

2041.2 

2214.8 

2395.6 

2583.4 

2778.3 

2980.3 

3189.4 

20 

1973.3 

2148.6 

2321.4 

2521.7 

2719.4 

2924.5 

3137.2 

3357.3 

LOGARITHMS. 

Logarithms  (abbreviation  log).  —  The  log  of  a  number  is  the  exponent 
of  the  power  to  which  it  is  necessary  to  raise  a  fixed  number  to  produce 
the  given  number.  The  fixed  number  is  called  the  base.  Thus  if  the 
base  is  10,  the  log  of  1000  is  3,  for  103  =  1000.  There  are  two  systems 
of  logs  in  general  use,  the  common,  in  which  the  base  is  10,  and  the  Naperian, 
or  hyperbolic,  in  which  the  base  is  2.718281828  ....  The  Naperian  base 
is  commonly  denoted  by  e,  as  in  the  equation  ey  —  x,  in  which  y  is  the 
Nap.  log  of  a:.  The  abbreviation  loge  is  commonly  used  to  denote  the 
Nap  log. 

In  any  system  of  logs,  the  log  of  1  is  0;  the  log  of  the  base,  taken  in  that 
system,  is  1.  In  any  system  the  base  of  which  is  greater  than  1,  the  logs  of 
all  numbers  greater  than  1  are  positive  and  the  logs  of  all  numbers  less 
than  1  are  negative. 

The  modulus  of  any  system  is  equal  to  the  reciprocal  of  the  Naperian  log 
of  the  base  of  that  system.  The  modulus  of  the  Naperian  system  is  1 ,  that 
of  the  common  system  is  0.4342945. 

The  log  of  a  number  in  any  system  equals  the  modulus  of  that  system  X 
the  Naperian  log  of  the  number. 

The  hyperbolic  or  Naperian  log  of  any  number  equals  the  common 
logX  2.3025851. 

Every  log  consists  of  two  parts,  an  entire  part  called  the  characteristic. 
or  index,  and  the  decimal  part,  or  mantissa.  The  mantissa  only  is  given 
in  the  usual  tables  of  common  logs,  with  the  decimal  point  omitted.  The 
characteristic  is  found  by  a  simple  rule,  viz.,  it  is  one  less  than  the  number 
of  figures  to  the  left  of  the  decimal  point  in  the  number  whose  log  is  to  be 
found.  Thus  the  characteristic  of  numbers  from  1  to  9.99  +  is  0,  from 
10  to  99.99  4-  is  1,  from  100  to  999  -f  is  2,  from  0.1  to  0.99  +  is  -  1,  from 
0.01  to  0.099  +  is  -2,  etc.  Thus 


log  of    2000  is  3.30103;  log  of  0.2 
••     "     oon   "  2.30103;     "    "  0.02 


200         , 

20  "  1.30103; 

2  "  Q.30103; 


is  -  1.30103,  or  9.30103  -  10 
"  -  2.30103,  "  8.30103  -  10 
0.002  "  -  3.30103,  "  7.30103  -  10 
1*  0,0002  "  -  4,30103,  '!  Q.301Q3  -  IQ 


136  LOdARITHMS   OF    NUMBERS. 

The  minus  sign  is  frequently  written  above  the  characteristic  thusi 
log  0.002  =  3.30103.  The  characteristic  only  is  negative,  the  decimal  part, 
or  mantissa,  being  always  positive. 

When  a  log  consists  of  a  negative  index  and  a  positive  mantissa,  it  is 
usual  to  write  the  negative  sign  over  the  index,  or  else  to  add  10  to  the 
index,  and  to  indicate  the  subtraction  of  10  from  the  resulting  logarithm. 

Thus  log  0.2  =  1.30103,  and  this  may  be  written  9.30103  -  10. 

In  tables  of  logarithmic  sines,  etc.,  the  —  10  is  generally  omitted,  as 
being  understood. 

Rules  for  use  of  the  table  of  logarithms.  — To  find  the  log  of  any 
whole  number.  —  For  1  to  100  inclusive  the  log  is  given  complete  in  the 
small  table  on  page  137. 

For  100  to  999  inclusive  the  decimal  part  of  the  log  is  given  opposite  the 
given  number  in  the  column  headed  0  in  the  table  (including  the  two 
figures  to  the  left,  making  six  figures).  Prefix  the  characteristic,  or 
index,  2. 

For  1000  to  9999  inclusive:  The  last  four  figures  of  the  log  are  found 
•  opposite  the  first  three  figures  of  the  given  number  and  in  the  vertical 
column  headed  with  the  fourth  figure  of  the  given  number ;  prefix  the  two 
figures  under  column  0,  and  the  index,  which  is  3. 

For  numbers  over  10,000  having  five  or  more  digits:  Find  the  decimal 
part  of  the  log  for  the  first  four  digits  as  above,  multiply  the  difference 
figure  in  the  last  column  by  the  remaining  digit  or  digits,  and  divide  by  10 
if  there  be  only  one  digit  more,  by  100  if  there  be  two  more,  and  so  on; 
add  the  quotient  to  the  log  of  the  first  four  digits  and  prefix  the  index, 
which  is  4  if  there  are  five  digits,  5  if  there  are  six  digits,  and  so  on.  The 
table  of  proportional  parts  may  be  used,  as  shown  below. 

To  find  the  log  of  a  decimal  fraction  or  of  a  whole  number  and  a 
'decimal.  —  First  find  the  log  of  the  quantity  as  if  there  were  no  decimal 
Doint,  then  prefix  the  index  according  to  rule:  the  index  is  one  less  than 
the  number  of  figures  to  the  left  of  the  decimal  point. 

Example,     log  of  3.14159.     log  of  3.141      =0.497068.    Diff.  =-138 
From  proportional  parts  5    =  690 

09=  1242 


log  3. 14159       0.4971494 

If  the  number  is  a  decimal  less  than  unity,  the  index  is  negative 
and  is  one  more  than  the_  number  of  zeros  to  the  right  of  the  decimal 
point.  Log  of  0.0682  =  2.833784  =  8.833784  -  10. 

To  find  the  number  corresponding  to  a  given  log. —  Find  in  the 
table  the  log  nearest  to  tne  decimal  part  of  the  given  log  and  take  the 
first  four  digits  of  the  required  number  from  the  column  N  and  the  top  or 
foot  of  the  column  containing  the  log  which  is  the  next  less  thanthegiven 
log.  To  find  the  5th  and  6th  digits  subtract  the  log  in  the  table  from  the 
given  log,  multiply  the  difference  by  100,  and  divide  by  the  figure  in  the 
Diff.  column  opposite  the  log;  annex  the  quotient  to  the  four  digits 
already  found,  and  place  the  decimal  point  according  to  the  rule;  the 
number  of  figures  to  the  left  of  the  decimal  point  is  one  greater  than  the 
index.  The  number  corresponding  to  a  log  is  called  the  anti-logarithm. 

Find  the  anti-log  of 0.497150 

Next  lowest  log  in  table  corresponds  to  3141 0.497068      Diff.  =  82 

Tabular  diff.  =  138;  82  -f-  138  =  0.59  -f- 
The  index  being  0,  the  number  is  therefore  3.14159  -f . 

To  multiply  two  numbers  by  tlie  use  of  logarithms.  —  Add  together 
the  logs  of  the  two  numbers,  and  find' the  number  whose  log  is  the  sum. 

To  divide  two  numbers.  —  Subtract  the  log  of  the  divisor  from  the 
log  of  the  dividend,  and  find  the  number  whose  log  is  the  difference. 
Log  of  a  fraction.     Log  of  a/b  =  log  a  —  log  b. 

To  raise  a  number  to  any  given  power.  —  Multiply  the  log  of  the 
number  by  the  exponent  of  the  power,  and  find  the  number  whose  log 
is  the  product. 

To  find  any  root  of  a  given  number.  —  Divide  the  log  of  the  number 
index  of  the  root.    The  quotient  is  tlje  log,  of  tfce  root. 


IiOGAUITHMS   OP  NUMBERS. 


137 


To  find  the  reciprocal  of  a  number.  —  Subtract  the  decimal  pait 
of  the  log  of  the  number  from  0,  add  1  to  the  index  and  change  the  sign  of 
the  index.  The  result  is  the  log  of  the  reciprocal. 

Required  the  reciprocal  of  3.141593. 

Log  of  3.141593,  as  found  above 0.4971498 

Subtract  decimal  part  from  0  gives 0.5028502 

Add  1  to  the  index,  and  changing  sign  of  the  index  gives. .  1. 5028502 
which  is  the  log  of  0.31831. 

To  find  the  fourth  term  of  a  proportion  by  logarithms.  —  Add 
the  logarithms  of  the  second  and  third  terms,  and  from  their  sum  subtract 
the  logarithm  of  the  first  term. 

When  one  logaithm  is  to  be  subtracted  from  another,  it  may  be  more 
convenient  to  convert  the  subtraction  into  an  addition,  which  may  be 
done  by  first  subtracting  the  given  logarithm  from  10,  adding  the  difference 
to  the  other  logarithm,  and  afterwards  rejecting  the  10. 

The  difference  between  a  given  logarithm  and  .10  is  called  its  arithmetical 
complement,  or  cologarithm. 

To  subtract  one  logarithm  from  another  is  the  same  as  to  add  its  com- 
plement and  then  reject  10  from  the  result.  For  a  —  b  =  10  —  b+  a  — 10. 

To  work  a  proportion,  then,  by  logarithms,  add  the  complement  of  the 
logarithm  of  the  first  term  to  the  logarithms  of  the  second  and  third  terms. 
The  characteristic  must  afterwards  be  diminished  by  10. 

Example     in    logarithms    with  a    negative    index.  —  Solve    by 


logarithms 


Vioii 
quotient  to  the  2.45  power. 

log  526    =       2.720986 
log  1011    =       3.004751 


which  means  divide  526  by  1011  and  raise  the 


log  of  quotient  = 
Multiply  by 


9.716235  -  10 
2.45 


.48581175 
3.8864940 
19.432470 


23. 80477575  -(10X2.45)  =  1.30477575  =  0.20173,  Ans. 
LOGARITHMS  OF  NUMBERS  FROM  1  TO  100. 


N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

! 

0.000000 

21 

.322219 

41 

.612784 

61 

.785330 

81 

.908485 

2 

0.301030 

22 

.342423 

42 

.623249 

62 

.792392 

82 

.913814 

0.477121 

23 

.361728 

43 

.633468 

63 

.799341 

83 

.919078 

4 

0.602060 

24 

.380211 

44 

.643453 

64 

.806180 

84 

.924279 

0.698970 

25 

.397940 

45 

.653213 

65 

.812913 

85 

.929419 

6 

0.778151 

26 

.414973 

46 

.662758 

66 

.819544 

86 

.934498 

7 

0.845098 

27 

.431364 

47 

.672098 

67 

.826075 

87 

.939519 

8 

0.903090 

28 

.447158 

48 

.681241 

68 

.832509 

88 

.944483 

9 

0.954243 

29 

.462398 

49 

.690196 

69 

.838849 

89 

.949390 

10 

1  .000000 

30 

.477121 

50 

.698970 

70 

.845098 

90 

.954243 

11 

.041393 

31 

.491362 

51 

.707570 

71 

.851258 

91 

.959041 

12 

.079181 

32 

.505150 

52 

.716003 

72 

.857332 

92 

.963788 

13 

.113943 

33 

.518514 

53 

.724276 

73 

.863323 

93 

.968483 

14 

.146128 

34 

.531479 

54 

.732394 

74 

.869232 

94 

.973128 

15 

.176091 

35 

.544068 

55 

.740363 

75 

.875061 

95 

977724 

16 

.204120 

36 

.556303 

56 

.748188 

76 

.880814 

96 

.982271 

17 

230449 

37 

.568202 

57 

.755875 

77 

.886491 

97 

.966772 

18 

.255273 

38 

.579784 

58 

.763428 

78 

.892095 

98 

.991226 

19 

.278754 

39 

.591065 

59 

.770852 

79 

.897627 

99 

.995635 

20 

1.301030 

40 

.602060 

60 

.778151 

80 

.903090 

100 

2.000000 

For  four-place  logarithms  see  page 


138 


LOGARITHMS   OF  NUMBERS. 


No.  100  L.  OOO.j 


[No.  109  L.  040. 


N. 

0 

1 

3 

0868 
5181 
9451 

3 

4 

5 

6 

7 

8 

346~1 

7748 

9 

3891 
8174 

Diff. 

432' 
428 

424 
420 

416 
412 
408 

404 
400 

397 

100 
1 
2 

3 

4 

5 
6 

8 
9 

000000 
4321 
8600 

0434 
4751 
9026 

1301 
5609 
9876 

1734 
6038 

2166 
6466 

2598 
6894 

3029 
7321 

0300 

4521 
8700 

0724 
4940 
9116 

1147 
5360 
9532 

1570 
5779 
9947 

1993 
6197 

2415 
6616 

012837 
7033 

3259 
7451 

3680 
7868 

4100 

8284 

0361 

4486 
8571 

2619 
6629 

0775 

4896 
8978 

021189 
5306 
9384 

1603 
5715 
9789 

2016 
6125 

2428 
6533 

2841 
6942 

3252 
7350 

3664 
7757 

4075 
8164 

0195 
4227 
8223 

0600 
4628 
8620 

1004 
5029 
9017 

1408 
5430 
9414 

1812 
5830 
9811 

2216 
6230 

3021 

7028 

033424 
7426 
04 

3826 
7825 

0207 

0602 

0998 

PROPORTIONAL  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

434" 

43.4 

86.8 

130.2 

173.6 

217.0 

260.4 

303.8 

347.2 

390.6 

433 

43.3 

86.6 

129.9 

173.2 

216.5 

259.8 

303.1 

346.4 

389.7 

432 

43.2 

86.4 

129.6 

172.8 

216.0 

259.2 

302.4 

345.6 

388.8 

431 

43.1 

86.2 

129.3 

172.4 

215.5 

258.6 

301.7 

344.8 

387.9 

430 

43.0 

86.0 

129.0 

172.0 

215.0 

258.0 

301.0 

344.0 

387.0 

429 

42.9 

85.8 

128.7 

171.6 

214.5 

257.4 

300.3 

343.2 

386.1 

428 

42.8 

85.6 

128.4 

171.2 

214.0 

256.8 

299.6 

342.4 

385.2 

427 

42.7 

85.4 

128.1 

170.8 

213.5 

256.2 

298.9 

341.6 

384.3 

426 

42.6 

85.2 

127.8 

170.4 

213.0 

255.6 

298.2 

340.8 

383.4 

425 

42.5 

85.0 

127.5 

170.0 

212.5 

255.0 

297.5 

340.0 

382.5 

424 

42.4 

84.8 

127.2 

169.6 

212.0 

254.4 

296.8 

339.2 

381.6 

423 

42.3 

84.6 

126.9 

169.2 

211.5 

253.8 

296.1 

338.4 

380.7 

422 

42.2 

84.4 

126.6 

168.8 

211.0 

253.2 

295.4 

337.6 

379.8 

421 

42.1 

84.2 

126.3 

168.4 

210.5 

252.6 

294.7 

336.8 

378.9 

420 

42.0 

84.0 

126.0 

168.0 

210.0 

252.0 

294.0 

336.0 

373.0 

419 

41.9 

83.8 

125.7 

167.6 

209.5 

251.4 

293.3 

335.2 

377.1 

418 

41.8 

83.6 

125.4 

167.2 

209.0 

250.8 

292.6 

334.4 

376.2 

417 

41.7 

83.4 

125.1 

166.8 

208.5 

250.2 

291.9 

333.6 

375.3 

416 

41.6 

83.2 

124.8 

166.4 

208.0 

249.6 

291.2 

332.8 

374.4 

415 

41.5 

83.0 

124.5 

166.0 

207.5 

249.0 

290.5 

332.0 

373.5 

414 

41.4 

82.8 

124.2 

165.6 

207.0 

248.4 

289.8 

331.2 

372.6 

413 

41.3 

82.6 

123.9 

165.2 

206.5 

247.8 

289.1 

330.4 

371.7 

412 

41.2 

82.4 

123.6 

164.8 

206.0 

247.2 

288.4 

329.6 

370.8 

>11 

41.1 

82.2 

123.3 

164.4 

205.5 

246.6 

287.7 

328.8 

369.9 

410 

41.0 

82.0 

123.0 

164.0 

205.0 

246.0 

287.0 

328.0 

369.0 

409 

40.9 

81.8 

122.7 

163.6 

204.5 

245.4 

286.3 

327.2 

368.1 

408 

40.8 

81.6 

122.4 

163.2 

204.0 

244.8 

285.6 

326.4 

367.2 

407 

40.7 

81.4 

122.1 

162.8 

203.5 

244.2 

284.9 

325.6 

366.3 

406 

40.6 

81.2 

121.8 

162.4 

203.0 

243.6 

284.2 

324.8 

365.4 

405 

40.5 

81.0 

121.5 

162.0 

202.5 

243.0 

283.5 

324.0 

364.5 

404 

40.4 

80.8 

121.2 

161.6 

202.0 

242.4 

282.8 

323.2 

363.6 

403 

40.3 

80.6 

120.9 

161.2 

201.5 

241.8 

282.1 

322.4 

362.7 

402 

40.2 

80.4 

120.6 

160.8 

201.0 

241.2 

281.4 

321.6 

361.8 

401 

40.1 

80.2 

120.3 

160.4 

200.5 

240.6 

280.7 

320.8 

360.9 

400 

40.0 

80.0 

120.0 

160.0 

200.0 

240.0 

280.0 

320.0 

360.0 

399 

39.9 

79.8 

119.7 

159.6 

199.5 

239.4 

279.3 

319.2 

359.1 

398 

39.8 

79.6 

119.4 

159.2 

199.0 

238.8 

278.6 

318.4 

358.2 

397 

39.7 

79.4 

119.1 

158.8 

198.5 

238.2 

277.9 

317.6 

357.3 

396 

39.6 

79.2 

118.8 

158.4 

198.0 

237.6 

277.2 

316.8 

356.4 

395 

39.5 

79.0 

118.5 

158.0 

197.5 

237.0 

276.5 

316.0 

355.5 

LOGARITHMS   OF   NUMBERS. 


139 


No.  110  L.  041.] 


[No.  119  L.  078. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

no 
i 

2 

3 
4 

5 

6 
7 

8 
9 

041393 
5323 
9218 

1787 
5714 
9606 

2182 
6105 
9993 

2576 
6495 

0380 
4230 
8046 

2969 
6885 

0766 
4613 
8426 

3362 
7275 

3755 
7664 

4148 
8053 

4540 
8442 

4932 
8830 

393 
390 

386 
383 

379 
376 
373 

370 
366 
363 

1153 
4996 
8805 

1538 
5378 
9185 

1924 
5760 
9563 

2309 
6142 
9942 

2694 
6524 

0320 
4083 
7815 

053078 
6905 

3463 
7286 

3846 
7666 

060698 
4458 
8186 

1075 
4832 
8557 

1452 
5206 
8928 

1829 
5580 
9298 

2206 
5953 
9668 

2582 
6326 

2958 
6699 

3333 
7071 

3709 
7443 

0038 
3718 
7368 

0407 
4085 
7731 

0776 
4451 
8094 

1145 
4816 
8457 

1514 
5182 
8819 

071882 
5547 

2250 
5912 

2617 
6276 

2985 
6640 

3352 
7004 

PROPORTIONAL   PA.RTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

395 

39.5 

79.0 

118.5 

158.0 

197.5 

237.0 

276.5 

316.0 

355.5 

394 

39.4 

78.8 

T18.2 

157.6 

197.0 

236.4 

275.8 

315.2 

354.6 

393 

39.3 

78.6 

1  17.9 

157.2 

196.5 

235.8 

275.1 

314.4 

353.7 

392 

39.2 

78.4 

117.6 

156.8 

196.0 

235.2 

274.4 

313.6 

352.8 

391 

39.1 

78.2 

117.3 

156.4 

195.5 

234.6 

273.7 

312.8 

351.9 

390 

39.0 

78.0 

117.0 

156.0 

195.0 

234.0 

273.0 

312.0 

351.C 

389 

38.9 

77.8 

116.7 

155.6 

194.5 

233.4 

272.3 

311.2 

350.1 

388 

38.8 

77.6 

116.4 

155.2 

194.0 

232.8 

271.6 

310.4 

349.2 

387 

38.7 

77.4 

116.1 

154.8 

193.5 

232.2 

270.9 

309.6 

348.3 

386 

38.6 

77.2 

115.8 

154.4 

193.0 

231.6 

270.2 

308.8 

347.4 

385 

38.5 

77.0 

115.5 

154.0 

192.5 

231.0 

269.5 

308.0 

346.? 

384 

38.4 

76.8 

115.2 

153.6 

192.0 

230.4 

268.8 

307.2 

345.6 

383 

38.3 

76.6 

114.9 

153.2 

191.5 

229.8 

268.1 

306.4 

344.7 

382 

38.2 

76.4 

114.6 

152.8 

191.0 

229.2 

267.4 

305.6 

343.8 

381 

38.1 

76.2 

114.3 

152.4 

190.5 

228.6 

266.7 

304.8 

342.9 

380 

38.0 

76.0 

114.0 

152.0 

1900 

228.0 

266.0 

304.0 

342.0 

379 

37.9 

75.8 

113.7 

151.6 

189.5 

227.4 

265.3 

303.2 

341.1 

378 

37.8 

75.6 

113.4 

151.2 

189.0 

226.8 

264.6 

302.4 

340.2 

377 

37.7 

75.4 

113.1 

150.8 

188.5 

226.2 

263.9 

301.6 

339.3 

376 

37.6 

75.2 

112.8 

150.4 

188.0 

225.6 

263.2 

300.8 

338.4 

375 

37.5 

75.0 

112.5 

150.0 

187.5 

225.0 

262.5 

300.0 

337.5 

374 

37.4 

74.8 

112.2 

149.6 

187.0 

224.4 

261.8 

299.2 

336.6 

373 

37.3 

74.6 

111.9 

149.2 

186.5 

223.8 

261.1 

298.4 

335.7 

372 

37.2 

74.4 

111.6 

148.8 

186.0 

223.2 

260.4 

297.6 

334.8 

371 

37.1 

74.2 

111.3 

148.4 

185.5 

222.6 

259.7 

296.8 

333.9 

370 

37.0 

74.0 

111.0 

148.0 

185.0 

222.0 

259.0 

296.0 

333.0 

369 

36.9 

73.8 

110.7 

147.6 

184.5 

221.4 

258.3 

295.2 

332.1 

368 

36.8 

73.6 

110.4 

147.2 

184.0 

220.8 

257.6 

294.4 

331.2 

367 

36.7 

73.4 

110.1 

146.8 

183.5 

220.2 

256.9 

293.6 

330.3 

366 

36.6 

73.2 

109.8 

146.4 

183.0 

219.6 

256.2 

292.8 

329.4 

365 

36.5 

73.0 

109.5 

146.0 

182.5 

219.0 

255.5 

292.0 

328.5 

364 

36.4 

72.8 

109.2 

145.6 

182.0 

218.4 

254.8 

291.2 

327.6 

363 

36.3 

72.6 

108.9 

145.2 

181.5 

217.8 

254.1 

290.4 

326.7 

362 

36.2 

72.4 

108.6 

144.8 

181.0 

217.2 

253.4 

289.6 

325.8 

361 

36.1 

72.2 

108.3 

144.4 

180.5 

216.6 

252.7 

288.8 

324.9 

360 

36.0 

72.0 

108.0 

144.0 

180.0 

216.0 

252.0 

288.0 

324.0 

359 

35.9 

71.8 

107.7 

143.6 

179.5 

215.4 

251.3 

287.2 

323.1 

358 

35.8 

71.6 

107.4 

143.2 

179.0 

214.8 

250.6 

286.4 

322.2 

357 

35.7 

71.4 

107.1 

142.8 

178.5 

214.2 

249.9 

285.6 

32!.  3 

356 

35.6 

71.2 

106.8 

142.4 

178.0 

213.6 

249.2 

284.8 

320.4 

140 


LOGARITHMS    OF   NUMBERS. 


No.  120  L.  079.] 


[No.  134  L.  130. 


N. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

Diff. 

120 
2 

4 

5 

6 
7 

8 

9 

130 

1 

2 
3 

4 

079181 

9543 

9904 

0266 
3861 
7426 

0626 
4219 
7781 

0987 
4576 
8136 

1347 
4934 
8490 

1707 
5291 
8845 

2067 
5647 
9198 

2426 
6004 
9552 

360 
357 
355 

352 
349 

346 
343 
341 

338 
335 

333 

330 
328 
325 

323 

082785 
6360 
9905 

3144 
6716 

3503 
7071 

0258 

3772 
7257 

0611 

4122 
7604 

0963 

4471 
7951 

1315 
4820 
8298 

1667 
5169 
8644 

2018 
5518 
8990 

2370 
5866 
9335 

2721 
6215 
9681 

3071 
6562 

093422 
6910 

0026 
3462 
6871 

100371 
3804 
7210 

0715 
4146 
7549 

1059 
4487 
7888 

1403 
4828 
8227 

1747 
5169 
8565 

2091 
5510 
8903 

2434 
5851 
9241 

2777 
6191 
9579 

3119 
6531 
9916 

0253 
3609 

6940 

110590 

3943 
725  '1 

0926 

4277 
7603 

1263 

4611 
7934 

1599 

4944 
8265 

1934 

5278 
8595 

2270 

5611 

8926 

2605 

5943 
9256 

2940 

6276 
9586 

3275 

6608 
9915 

0245 
3525 
6781 

120574 
3852 
7105 
13 

0903 
4178 
7429 

1231 
4504 
7753 

1560 
4830 
8076 

1888 
5156 
8399 

2216 
5481 
8722 

2544 
5806 
9045 

2871 
6131 
9368 

3198 
6456 
9690 

0012 

PROPORTIONAL  PARTS. 


Diff. 

1 

3 

3 

4 

5 

6 

7 

8 

9 

355" 

35.5 

71.0 

106.5 

142.0 

177.5 

213.0 

248.5 

284.0 

319.5 

354 

35.4 

70.8 

106.2 

141.6 

177.0 

212.4 

247.8 

283.2 

318.6 

353 

35.3 

70.6 

105.9 

141.2 

176.5 

211.8 

247.1 

282.4 

317.7 

352 

35.2 

70.4 

105.6 

140.8 

176.0 

211.2 

246.4 

281.6 

316.8 

351 

35.1 

70.2 

105.3 

140.4 

175.5 

210.6 

245.7 

280.8 

315.9 

350 

35.0 

70.0 

105.0 

140.0 

175.0 

210.0 

245.0 

280.0 

315.0 

349 

34.9 

69.8 

104.7 

139.6 

174.5 

209.4 

244.3 

279.2 

314.1 

348 

34.8 

69.6 

104.4 

139.2 

174.0 

208.8 

243.6 

278.4 

313.2 

347 

34.7 

694 

104.1 

138.8 

173.5 

208.2 

242.9 

277.6 

312.3 

346 

34.6 

69.2 

103.8 

138.4 

173.0 

207.6 

242.2 

276.8 

311  4 

345 

34.5 

69.0 

103.5 

138.0 

172.5 

207.0 

241.5 

276.0 

310.5 

344 

34.4 

68.8 

103.2 

137.6 

172.0 

206.4 

240.8 

275.2 

309.6 

343 

34.3 

68.6 

102.9 

137.2 

171.5 

205.8 

240.1 

274.4 

308.7 

342 

34.2 

68.4 

102.6 

136.8 

171.0 

205.2 

239.4 

273.6 

307.8 

341 

34.1 

68.2 

102.3 

136.4 

170.5 

204.6 

238.7 

272.8 

306.9 

340 

34.0 

68.0 

102.0 

136.0 

170.0 

204.0 

238.0 

272.0 

306.0 

339 

33.9 

67.8 

101.7 

135.6 

169.5 

203.4 

237.3 

271.2 

305.1 

338 

33.8 

67.6 

101.4 

135.2 

169.0 

202.8 

236.6 

270.4 

304.2 

337 

33.7 

67.4 

101.1 

134.8 

168.5 

202.2 

235.9 

269.6 

303.3 

336 

33.6 

67.2 

100.8 

134.4 

168.0 

201.6 

235.2 

268.8 

302.4 

335 

33.5 

67.0 

100.5 

134.0 

167.5 

201.0 

234.5 

268.0 

301.5 

334 

33.4 

66.8 

100.2 

133.6 

167.0 

200.4 

233.8 

267.2 

300.6 

333 

33.3 

66.6 

99.9 

133.2 

166.5 

199.8 

233.1 

266.4 

299.7 

332 

33.2 

66.4 

99.6 

132.8 

166.0 

199.2 

232.4 

265.6 

298.8 

331 

33.1 

66.2 

99.3 

132.4 

165.5 

198.6 

231.7 

264.8 

297.9 

330 

33.0 

66.0 

99.0 

132.0 

165.0 

198.0 

231.0 

264.0 

297.0 

329 

32.9 

65.8 

98.7 

131.6 

164.5 

197.4 

230.3 

263.2 

296.1 

328 

32.8 

65.6 

98.4 

131.2 

164.0 

196.8 

229.6 

262.4 

295.2 

327 

32.7 

65.4 

98.1 

130.8 

163.5 

196.2 

228.9 

261.6 

294.3 

326 

32.6. 

65.2 

97.8 

130.4 

163.0 

195.6 

228.2 

260.8 

293.4 

325 

32.5 

65.0 

97.5 

130.0 

162.5 

195.0 

227.5 

260.0 

292.5 

324 

32.4 

64.8 

97.2 

129.6 

162.0 

194.4 

226.8 

259.2 

291.6 

323 

32.3 

64.6 

96.9 

129.2 

161.5 

193.8 

226.1 

258.4 

290.7 

322 

32.2 

64.4 

96.6 

128.8 

161.0 

193.2 

225.4 

257.6 

289.8 

LOGARITHMS    OF  NUMBERS. 


141 


No.  135  L.  130.] 


[No.  149  L.  175. 


N. 

O 

1 

3 

3 

4 

5 

6 

7 

8 

9 

Diff. 

~32T 
318 
316 

314 
311 

309 

307 
305 
303 

301 
299 
297 
295 

293 
291 

135 
6 

7 
8 

9 
140 

2 
3 
4 

5 
6 

7 

8 
9 

130334 
3539 
6721 
9879 

0655 
3858 
7037 

0977 
4177 
7354 

1298 
4496 
7671 

1619 
4814 
7987 

1939 
5133 
8303 

2260 
5451 
8618 

2580 
5769 
8934 

2900 
6086 
9249 

3219 
6403 
9564 

0194 
3327 

6438 
9527 

0508 
3639 

6748 
9835 

0822 
3951 

7058 

1136 
4263 

7367 

1450 
4574 

7676 

1763 
4885 

7985 

2076 
5196 

8294 

2389 
5507 

8603 

2702 
5818 

8911 

143015 

6128 
9219 

0142 
3205 
6246 
9266 

0449 
3510 
6549 
9567 

0756 
3815 
6852 
9868 

1063 
4120 
7154 

1370 
4424 
7457 

1676 
4728 
7759 

1982 
5032 
8061 

152288 
5336 
8362 

2594 
5640 
8664 

2900 
5943 
8965 

0168 
3161 
6134 
9086 

0469 
3460 
6430 
9380 

0769 
3758 
6726 
9674 

1068 
4055 
7022 
9968 

161368 
4353 
7317 

1667 
4650 
7613 

1967 
4947 
7908 

2266 
5244 
8203 

2564 

5541 
8497 

2863 
5838 
8792 

1  70262 
3186 

0555 

3478 

0348 
3769 

1141 
4060 

1434 
4351 

1726 
4641 

2019 
4932 

2311 
5222 

2603 
5512 

2895 
5802 

PROPORTIONAL  PARTS. 


Diff. 

1 

3 

3 

4 

5 

6 

7 

8 

9 

321 

32.1 

64.2 

96.3 

128.4 

160.5 

192.6 

224.7 

256.8 

288.9 

320 

32.0 

64.0 

96.0 

128.0 

160.0 

192.0 

224.0 

256.0 

2880 

319 

31.9 

63.8 

95.7 

127.6 

159.5 

191.4 

223.3 

255.2 

287.1 

318 

31.8 

63.6 

95.4 

127.2 

159.0 

190.8 

222.6 

254.4 

286.2 

317 

31.7 

63.4 

95.1 

126.8 

158.5 

190.2 

221.9 

253.6 

285.3 

316 

31.6 

63.2 

94.8 

126.4 

158.0 

189.6 

221.2 

252.8 

284.4 

315 

31.5 

63.0 

94.5 

126.0 

157.5 

189.0 

220.5 

252.0 

283.5 

314 

31.4 

62.8 

94.2 

125.6 

157.0 

188.4 

219.8 

251.2 

282.6 

313 

31.3 

62.6 

93.9 

125.2 

156.5 

187.8 

219.1 

250.4 

281.7 

312 

31.2 

62.4 

93.6 

124.8 

156.0 

187.2 

218.4 

249.6 

280.8 

311 

31.1 

62.2 

93.3 

124.4 

155.5 

186.6 

217.7 

248.8 

279.9 

310 

31.0 

62.0 

93.0 

124.0 

155.0 

186.0 

217.0 

248.0 

279.0 

309 

30.9 

61.8 

92.7 

123.6 

154.5 

185.4 

216.3 

247.2 

278.1 

308 

30.8 

61.6 

92.4 

123.2 

154.0 

184.8 

215.6 

246.4 

277.2 

307 

30.7 

61.4 

92.1 

122.8 

153.5 

184.2 

214.9 

245.6 

276.3 

306 

30.6 

61.2 

91.8 

122.4 

153.0 

183.6 

214.2 

244.8 

275.4 

305 

30!5 

61.0 

91.5 

122.0 

152  5 

183.0 

213.5 

244.0 

274.5 

304 

30.4 

60.8 

91.2 

121.6 

152.0 

182.4 

212.8 

243.2 

273.6 

303 

30.3 

60.6 

90.9 

121.2 

151.5 

181.8 

212.1 

242.4 

272.7 

302 

30.2 

60.4 

90.6 

120.8 

151.0 

181.2 

211.4 

241.6 

271.8 

301 

30.1 

60.2 

90.3 

120.4 

150.5 

1806 

210.7 

240.8 

270.9 

300 

30.0 

60.0 

90.0 

120.0 

150.0 

180.0 

210.0 

240.0 

270.0 

299 

29.9 

59.8 

89.7 

119.6 

149.5 

179.4 

209.3 

239.2 

269.1 

298 

29.8 

59.6 

89.4 

119.2 

149.0 

178.8 

208.6 

238.4 

268.2 

297 

29.7 

59.4 

89.1 

118.8 

148.5 

178.2 

207.9 

237.6 

267.3 

296 

29.6 

59.2 

88.8 

118.4 

148.0 

177.6 

207.2 

236.8 

266.4 

295 

29.5 

590 

88.5 

118.0 

147.5 

177.0 

206.5 

236.0 

265.5 

294 

29.4 

58.8 

88.2 

117.6 

147.0 

176.4 

205.8 

235.2 

264.6 

293 

29.3 

58.6 

87.9 

117.2 

146.5 

175.8 

205.1 

234.4 

263.7 

292 

29.2 

58.4 

87.6 

116.8 

146,0 

175.2 

204.4 

233.6 

262.8 

291 

29.1 

58.2 

87.3 

116.4 

145.5 

174.6 

203.7 

232.8 

261.9 

290 

29.0 

58.0 

87.0 

116.0 

145.0 

174.0 

203.0 

232.0 

261.. 

289 

28.9 

57.8 

86.7 

115.6 

144.5 

173.4 

202.3 

231.2 

260.1 

288 

28.8 

57.6 

86.4 

115.2 

144.0 

172.3 

201.6 

230.4 

259.2 

287 

28.7 

57.4 

86.1 

114.8 

143.5 

172.2 

200.9 

229.6 

258.3 

286 

28.6 

57.2 

85.8 

114.4 

143.0 

171.6 

200.2 

228.8 

2*7.4 

142 


LOGARITHMS   OP  NUMBERS. 


Wo.  150  L.  176.] 


[No.  109  L.  230 


N. 
~T50" 

2 

4 

5 
6 
7 

8 

9 
160 
2 

3 

4 

6 

8 
9 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

DiflF. 
-28T 

287 
285 

283 

281 
279 
278 
276 

274 
272 

271 
269 

267 
266 
264 
262 

261 
259 
258 

256 

176091 
8977 

6381 
9264 

6670 
9552 

6959 
9839 

7248 

7536 

7825 

8113 

8401 

8689 

0126 
2985 
5825 
8647 

0413 
3270 
6108 
8928 

0699 
3555 
6391 
9209 

0986 
3839 
6674 
9490 

1272 
4123 
6956 
9771 

1558 
4407 
7239 

181844 
4691 
7521 

2129 
4975 
7803 

2415 
5259 
8084 

2700 
5542 
8366 

0051 
2846 
5623 
8382 

190332 
3125 
5900 
8657 

0612 
3403 
6176 
8932 

0892 
3681 
6453 
9206 

1171 
3959 
6729 
9481 

1451 
4237 
7005 
9755 

1730 
4514 
7281 

2010 
4792 
7556 

2289 
5069 
7832 

2567 
5346 
8107 

0029 
2761 

5475 
8173 

0303 
3033 

5746 
8441 

0577 
3305 

6016 
8710 

0850 
3577 

6286 
8979 

1124 
3848 

6556 
9247 

201397 

4120 
6826 
9515 

1670 

4391 
7096 
9783 

1943 

4663 
7365 

2216 

4934 
7634 

2488 

5204 
7904 

0051 
2720 
5373 
8010 

0319 
2986 
5638 
8273 

0586 
3252 
5902 
8536 

0853 
3518 
6166 
8798 

1121 
3783 
6430 
9060 

1388 
4049 
6694 
9323 

1654 
4314 
6957 
9585 

1921 
4579 
7221 
9846 

212188 
4844 
7484 

2454 
5109 
7747 

220108 
2716 
5309 
7887 
23 

0370 
2976 
5568 
8144 

0631 
3236 
5826 
8400 

0892 
3496 
6084 
8657 

1153 
3755 
6342 
8913 

1414 
4015 
6600 
9170 

1675 
4274 
6858 
9426 

1936 
4533 
7115 
9682 

2196 
4792 
7372 
9938 

2456 
5051 
7630 

0193 

PROPORTIONAL  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

285 

28.5 

57.0 

85.5 

114.0 

142.5 

171.0 

199.5 

228.0 

256.5 

,784 

28.4 

56.8 

85.2 

113.6 

142.0 

170.4 

198.8 

227.2 

255.6 

283 

28.3 

56.6 

84.9 

113.2 

141.5 

169.8 

198.1 

226.4 

254.7 

282 

28.2 

56.4 

84.6 

112.8 

141.0 

169.2 

197.4 

225.6 

253.8 

281 

28.1 

56.2 

84.3 

112.4 

140.5 

168.6 

196.7 

224.8 

252.9 

280 

28.0 

56.0 

84.0 

112.0 

140.0 

168.0 

196.0 

224.0 

252.0 

279 

27.9 

55.8 

83.7 

111.6 

139.5 

167.4 

195.3 

223.2 

251.1 

278 

27.8 

55.6 

83.4 

111.2 

139.0 

166.8 

194.6 

222.4 

250.2 

277 

27.7 

55.4 

83.1 

110.8 

138.5 

166.2 

193.9 

221.6 

249.3 

276 

27.6 

55.2 

82.8 

110.4 

138.0 

165.6 

193.2 

220.8 

248.4 

275 

27.5 

55.0 

82.5 

110.0 

137.5 

165.0 

192.5 

220.0 

247.5 

274 

27.4 

54.8 

82.2 

109.6 

137.0 

164.4 

191.8 

219.2 

246.6 

273 

27.3 

54.6 

81.9 

109,2 

136.5 

163.8 

191.1 

218.4 

245.7 

272 

27.2 

54.4 

81.6 

108.8 

136.0 

163.2 

190.4 

217.6 

244.8 

271 

27.1 

54.2 

81.3 

108.4 

135.5 

162.6 

189.7 

216.8 

243.9 

270 

27.0 

54.0 

81.0 

108.0 

135.0 

162.0 

189.0 

216.0 

243.0 

269 

26.9 

53.8 

80.7 

107.6 

134.5 

161.4 

188.3 

215.2 

242.1 

268 

26.8 

53.6 

80.4 

107.2 

134.0 

160.8 

187.6 

214.4 

241.2 

267 

26.7 

53.4 

80.1 

106.8 

133.5 

160.2 

186.9 

213.6 

240.3 

266 

26.6 

53.2 

79.8 

106.4 

133.0 

159.6 

186.2 

212.8 

239.4 

265 

26.5 

53.0 

79.5 

106.0 

132.5 

159.0 

185.5 

212.0 

238.5 

264 

26.4 

52.8 

79.2 

105.6 

132.0 

158.4 

184.8 

211.2 

237.6 

263 

26.3 

52.6 

78.9 

105.2 

131.5 

157.8 

184.1 

210.4 

236.7 

262 

26.2 

52.4 

78.6 

104.8 

131.0 

157.2 

183.4 

209.6 

235.8 

261 

26.1 

52.2 

78.3 

104.4 

130.5 

156.6 

182.7 

208.8 

234.9 

260 

26.0 

52.0 

78.0 

104.0 

130.0 

156.0 

182.0 

208.0 

234.0 

259 

25.9 

51.8 

77.7 

103.6 

129.5 

155.4 

181.3 

207.2 

233.1 

258 

25.8 

51.6 

77.4 

103.2 

129.0 

154.8 

180.6 

206.4 

232.2 

257 

25.7 

51.4 

77.1 

102.8 

128.5 

154.2 

179.9 

205.6 

231.3 

256 

25.6 

51.2 

76,8 

102.4 

128.0 

153.6 

179.2 

204.8 

230.4 

255 

25.5 

51.0 

76.5  ' 

102.0 

127.5  ' 

153.0 

178.5 

204.0  i 

229J 

LOGARITHMS   OF  NUMBERS. 


143 


No.  I  VOL.  230.] 


[No.  189L.278. 


N. 

17o~ 

i 

2 
3 

4 

6 
7 

8 
9 

180 
1 

2 
3 

4 
5 
6 

8 
9 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

230449 
2996 
5528 
8046 

0704 
3250 
5781 
8297 

0960 
3504 
6033 
8548 

1215 
3757 
6285 
8799 

1470 
4011 
6537 
9049 

1724 
4264 
6789 
9299 

1979 
4517 
7041 
9550 

2234 
4770 
7292 
9800 

2488 
5023 
7544 

2742 
5276 
7795 

255 
253 
252 

250 

249 
248 
246 

245 
243 
242 

241 
239 

238 
237 
235 
234 

233 
232 
230 
229 

0050 
2541 
5019 
7482 
9932 

0300 
2790 
5266 
7728 

240549 
3038 
5513 
7973 

0799 
3286 
5759 
8219 

1048 
3534 
6006 
8464 

1297 
3782 
6252 
8709 

1546 
4030 
6499 
8954 

1795 
4277 
6745 
9198 

2044 
4525 
6991 
9443 

2293 
4772 
7237 
9687 

0176 
2610 
5031 

7439 
9833 

250420 
2853 

5273 
7679 

0664 
3096 

5514 
7918 

0908 
3338 

5755 
8158 

1151 
3580 

5996 
8398 

1395 

3822 

6237 
8637 

1638 
4064 

6477 
8877 

1881 
4306 

6718 
9116 

2125 
4548 

6958 
9355 

2368 
4790 

7198 
9594 

260071 
2451 
4818 
7172 
9513 

0310 
2688 
5054 
7406 
9746 

0548 
2925 
5290 
7641 
9980 

0787 
3162 
5525 
7875 

1025 
3399 
5761 
81  10 

1263 
3636 
5996 
8344 

1501 
3873 
6232 
8578 

1739 
4109 
6467 
8812 

1976 
4346 
6702 
9046 

2214 
4582 
6937 
9279 

0213 
2538 
4850 
7151 

0446 
2770 
5081 
7380 

0679 
3001 
5311 
•7609 

0912 
3233 
5542 
7838 

1144 
3464 
5772 
8067 

1377 
3696 
6002 
8296 

1609 
3927 
6232 
8525 

271842 
4158 
6462 

2074 
4389 
6692 

2306 
4620 
6921 

PROPORTIONAL  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

255 

25.5 

51.0 

76.5 

102.0 

127.5 

153.0 

178.5 

204.0 

229.5 

254 

25.4 

50.8 

76.2 

101.6 

127.0 

152.4 

177.8 

203.2 

228.6 

253 

25.3 

50.6 

75.9 

101.2 

126.5 

151.8 

177.1 

202.4 

227.7 

252 

25.2 

50.4 

75.6 

100.8 

126.0 

151.2 

176.4 

201.6 

226.8 

251 

25.1 

50.2 

75.3 

100.4 

125.5 

150.6 

175.7 

200.8 

225.9 

250 

25.0 

50.0 

75.0 

100.0 

125.0 

150.0 

175.0 

200.0 

225.0 

249 

24.9 

49.8 

74.7 

99.6 

124.5 

149.4 

174.3 

199.2 

224.1 

248 

24.8 

49.6 

74.4 

99.2 

124.0 

148.8 

173.6 

198.4 

223.2 

247 

24.7 

49.4 

74.1 

98.8 

123.5 

148.2 

172.9 

197.6 

222.3 

246 

24.6 

49.2 

73.8 

98.4 

123.0 

147.6 

172.2 

196.8 

221.4 

245 

24.5 

49.0 

73.5 

98.0 

122.5 

147.0 

171.5 

196.0 

220.5 

244 

24.4 

48.8 

73.2 

97.6 

122.0 

146.4 

170.8 

195.2 

219.6 

243 

24.3 

48.6 

72.9 

97.2 

121.5 

145.8 

170.1 

194.4 

218.7 

242 

24.2 

48.4 

72.6 

96.8 

121.0 

145.2 

1694 

193.6 

217.8 

241 

24.1 

48.2 

72.3 

96.4 

120.5 

144.6 

168.7 

192.8 

216.9 

240 

24.0 

48.0 

72.0 

96.0 

120.0 

144.0 

168.0 

192.0 

216.0 

239 

23.9 

47.8 

71.7 

95.6 

119.5 

143.4 

167.3 

191.2 

215.1 

238 

23.8 

47.6 

71.4 

95.2 

119.0 

142.8 

166.6 

190.4 

214.2 

237 

23.7 

47.4 

71.1 

94.8 

118.5 

142.2 

165.9 

189.6 

213.3 

236 

23.6 

47.2 

70.8 

94.4 

118.0 

141.6 

165.2 

188.8 

212.4 

235 

23.5 

47.0 

70.5 

94.0 

117.5 

141.0 

164.5 

188.0 

211.5 

234 

23.4 

46.8 

70.2 

93.6 

117.0 

140.4 

163.8 

187.2 

210.6 

233 

23.3 

46.6 

69.9 

93.2 

116.5 

139.8 

163.1 

186.4 

209.7 

232 

23.2 

46.4 

69.6 

92.8 

116.0 

139.2 

162.4 

185.6 

208.8 

231 

23.1 

46.2 

69.3 

92.4 

115.5 

138.6 

161.7 

184.8 

207.9 

230 

23.0 

46.0 

69.0 

920 

115.0 

138.0 

161.0 

184.0 

207.0 

229 

22.9 

45.8 

68.7 

91.6 

114.5 

137.4 

160.3 

183.2 

206.1 

228 

22.8 

45.6 

68.4 

91.2 

114.0 

136.8 

159.6 

182.4 

205.2 

227 

22.7 

45.4 

68.1 

90.8 

113.5 

136.2 

158.9 

181.6 

204.3 

226 

22.6 

45.2 

67.8 

90.4 

113.0 

135.6 

158.2 

180.8 

203.4 

144 


LOGARITHMS   OF  NUMBERS. 


No.  190  L.  278.] 


[No.  214  L.332. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

190 

278754 

898;; 

921  1 

9439 

9667 

9895 

m  j-i 

nae 

n^7« 

flQflA 

2 
3 
4 

281033 
3301 
5557 
7802 

1261 
3527 
5782 
8026 

1488 
3753 
6007 
8249 

1715 
3979 
6232 
8473 

1942 
4205 
6456 
8696 

2169 
4431 
6681 
8920 

2396 
4656 
6905 
9143 

2622 
4882 
7130 
9366 

2849 
5107 
7354 
9589 

3075 
5332 

7578 
9812 

227 
226 
225 
223 

5 
6 
7 
8 
9 

290035 
2256 
4466 
6665 
8853 

0257 
2478 
4687 
6884 
9071 

0480 
2699 
4907 
7104 
9289 

0702 
2920 
5127 
7323 
9507 

0925 
3141 
5347 
7542 
9725 

1147 
3363 
5567 
7761 
9943 

1369 
3584 
5787 
7979 

1591 
3804 
6007 
8198 

1813 
4025 
6226 
8416 

2034 
4246 
6446 
8635 

222 
221 
220 
219 

0161 

0-170 

ft^QS 

no  i  q 

9  t  A 

200 
1 
2 
3 
4 

301030 
3196 
5351 
7496 
9630 

1247 
3412 
5566 
7710 
9843 

1464 
3628 
5781 
7924 

1681 
3844 
5996 
8137 

1898 
4059 
6211 
8351 

2114 
4275 
6425 
8564 

2331 

4491 
6639 

8778 

2547 
4706 
6854 
8991 

2764 
4921 
7068 
9204 

2980 
5136 
7282 
9417 

217 
216 
215 
213 

0056 

0268 

048  1 

0693 

0906 

1  1  j  o 

i  -2ar\ 

1  ^49 

919 

5 
6 
7 
8 

311754 
3867 
5970 
8063 

1966 
4078 
6180 

8272 

2177 
4289 
6390 
8481 

2389 
4499 
6599 
8689 

2600 
4710 
6809 
8898 

2812 
4920 
7018 
9106 

3023 
5130 

7227 
9314 

3234 
5340 
7436 
9522 

3445 
5551 
7646 
9730 

3656 
5760 

7854 
9938 

211 
210 
209 
208 

9 
210 

2 
3 

320146 

2219 
4282 
6336 
8380 

0354 

2426 
4488 
6541 
8583 

0562 

2633 
4694 
6745 
8787 

0769 

2839 
4899 
6950 

8991 

t)977 

3046 
5105 
7155 
9194 

1184 

3252 
5310 
7359 
9398 

1391 

3458 
5516 
7563 
9601 

1598 

3665 
5721 
7767 
9805 

1805 

3871 
5926 
7972 

2012 

4077 
6131 
8176 

207 

206 
205 
204 

noftP. 

O9  i  i 

9flT 

4 

330414 

0617 

0819 

1022 

1225 

1427 

1630 

1832 

2034 

2236 

202 

PROPORTIONAL  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

225 

22.5 

45.0 

67.5 

90.0 

112.5 

135.0 

157.5 

180.0 

202.5 

224 

22.4 

44.8 

67.2 

89  6 

112.0 

134.4 

156.8 

179.2 

201.6 

223 

22.3 

44.6 

66.9 

89.2 

111.5 

133.8 

156.1 

178.4 

200.7 

222 

22.2 

44.4 

66.6 

88.8 

1110 

133.2 

155.4 

177.6 

199.8 

221 

22.1 

44.2 

66.3 

88.4 

110.5 

132.6 

154.7 

176.8 

198.9 

220 

22.0 

44.0 

66.0 

88.0 

110.0 

132.0 

154.0 

176.0 

198.0 

219 

21.9 

43.8 

65.7 

87.6 

109.5 

131.4 

153.3 

175.2 

197.1 

218 

21.8 

43.6 

65.4 

87.2 

109.0 

130.8 

152.6 

174.4 

196.2 

217 

21.7 

43.4 

65.1 

86.8 

108.5 

130.2 

151.9 

173.6 

195.3 

216 

21.6 

43.2 

64.8 

86.4 

108.0 

129.6 

151.2 

172.8 

194.4 

215 

21.5 

43.0 

64.5 

86.0 

107.5 

129.0 

150.5 

172.0 

193.5 

214 

21.4 

42.8 

64.2 

85.6 

107.0 

128.4 

149.8 

171.2 

192.6 

213 

21.3 

42.6 

63.9 

85.2 

106.5 

127.8 

149.1 

170.4 

191.7 

212 

21.2 

42.4 

63.6 

84.8 

106.0 

127.2 

148.4 

169.6 

190.8 

211 

21.1 

42.2 

63.3 

84.4 

105.5 

126.6 

147.7 

168.8 

189.9 

210 

21.0 

42.0 

63.0 

84.0 

105.0 

126.0 

147.0 

168.0 

189.0 

209 

20.9 

41.8 

62.7 

83.6 

104.5 

125.4 

146.3 

167.2 

188.1 

208 

20.8 

41.6 

62.4 

83.2 

104.0 

124.8 

145.6 

166.4 

187.2 

207 

20.7 

41.4 

62.1 

82.8 

103.5 

124.2 

144.9 

165.6 

186.3 

206 

20.6 

41:2 

61.8 

82.4 

103.0 

123.6 

144.2 

164.8 

185.4 

205 

20.5 

41.0 

61.5 

82.0 

102.5 

123.0 

143  5 

164.0 

184.5 

204 

20.4 

40.8 

61.2 

81.6 

102.0 

122.4 

142.8 

163.2 

183.6 

203 

20.3 

40.6 

60.9 

81.2 

101.5 

121.8 

142.1 

162.4 

182.7 

202 

20.2 

40.4 

60.6 

80.8 

101.0 

121.2   1 

141.4 

161.6 

181.8 

LOGARITHMS   OF   NUMBERS. 


145 


No.  215  L.  332.] 


[No.  239  L.  380. 


N. 

0 

1 

3 

3 

4 

5 

"3447 
5458 
7459 
9451 

6 

73649 
5658 
7659 
9650 

7 

~3850 
5859 
7858 
9849 

8 

~~4Q5~1 
6059 
8058 

9 

"4253 
6260 

S257 

Diff. 

~202" 
201 
200 

199 
198 

197 
196 
195 

194 
193 
193 
192 
191 
190 

189 

188 
188 
187 
186 

215 
6 

8 
9 
220 

2 
3 

4 
6 

8 
9 

230 

2 
3 
4 

5 
6 
7 
8 
9 

332438 
4454 
6460 
8456 

2640 
4655 
6660 
8656 

2842 
4856 
6860 
8855 

3044 
5057 
7060 
9054 

3246 
5257 
7260 
9253 

0047 
2028 

3999 
5962 
7915 
9860 

1796 
3724 
5643 
7554 
9456 

0246 
2225 

4195 
6157 
8110 

340444 

2423 
4392 
6353 
8305 

0642 

2620 
4589 
6549 
8500 

0841 

2817 
4785 
6744 
8694 

1039 

3014 
4981 
6939 
8889 

1237 

3212 
5178 
7135 
9083 

1435 

3409 
5374 
7330 
9278 

1632 

3606 
5570 
7525 
9472 

1830 

3802 
5766 
7720 
9666 

1603 
3532 
5452 
7363 
9266 

0054 
1989 
3916 
5834 
7744 
9646 

350248 
2183 
4108 
6026 
7935 
9835 

0442 
2375 
4301 
6217 
8125 

0636 
2568 
4493 
6408 
8316 

0829 
2761 
4685 
6599 
8506 

1023 
2954 
4876 
6790 
8696 

1216 
3147 
5068 
6981 
8886 

1410 
3339 
5260 
7172 
9076 

0025 

1917 
3800 
5675 
7542 
9401 

0215 

2105 
3988 
5862 
7729 
9587 

0404 

2294 
4176 
6049 
7915 
9772 

0593 

2482 
4363 
6236 
8101 
9958 

0783 

2671 
4551 
6423 

8287 

0972 

2859 
4739 
6610 
8473 

1161 

3048 
4926 
6796 
8659 

1350 

3236 
5113 
6983 
8845 

0698 
2544 
4382 
6212 
8034 
9849 

1539 

3424 
5301 
7169 
9030 

361728 
3612 
5488 
7356 
9216 

0143 
1991 
3831 
5664 
7488 
9306 

0328 
2175 
4015 
5846 
7670 
9487 

0513 
2360 
4198 
6029 
7852 
9668 

0883 
2728 
4565 
6394 
8216 

185 
184 
184 
183 
182 

181 

37106S 
2912 
4748 
6577 
8398 
38 

1253 
3096 
4932 
6759 
8580 

1437 
3280 
5115 
6942 
8761 

1622 
3464 
5298 
7124 
8943 

1806 
3647 
5481 
7306 
9124 

0030 

PROPORTIONAL  PARTS. 


Diff. 

1 

3 

3 

4 

5 

6 

7 

8 

9 

*202~ 
201 

20.2 
20.1 

40.4 
40.2 

60.6 
60.3 

80.8 
80.4 

101.0 
100.5 

121.2 
120.6 

141.4 
140.7 

161.6 
160.8 

181.8 
180.9 

200 

20.0 

40.0 

60.0 

80.0 

100.0 

120.0 

140.0 

160.0 

180.0 

199 

19.9 

39.8 

59.7 

79.6 

99.5 

119.4 

139.3 

159.2 

179.{ 

198 

19.8 

39.6 

59.4 

79.2 

99.0 

118.8 

138.6 

158.4 

178.2 

197 

19.7 

39.4 

59.1 

78.8 

98.5 

118.2 

137.9 

157.6 

177.3 

196 

19.6 

39.2 

58.8 

78.4 

98.0 

117.6 

137.2 

156.8 

176.4 

195 

19.5 

39.0 

58.5 

78.0 

97.5 

117.0 

136.5 

1  56.0 

175.5 

194 

19.4 

38.8 

58.2 

77.6 

97.0 

116.4 

135.8 

155.2 

174.6 

193 

19.3 

38.6 

57.9 

77.2 

96.5 

115.8 

135.1 

154.4 

173.7 

192 

19.2 

38.4 

57.6 

76.8 

96.0 

'  115.2 

134.4 

153.6 

172.8 

191 

19.1 

38.2 

57.3 

76.4 

95.5 

114.6 

133.7 

152.8 

171.9 

190 

190 

38.0 

57.0 

76.0 

95.0 

114.0 

133.0 

152.0 

171.0 

189 

18.9 

37.8 

56.7 

75.6 

94.5 

113.4 

132.3 

151.2 

170.1 

188 

18.8 

37.6 

56.4 

75.2 

940 

112.8 

131.6 

150.4 

169.2 

187 

18.7 

37.4 

56.1 

74.8 

93.5 

112.2 

130.9 

149.6 

168.3 

186 

18.6 

37.2 

55.3 

74.4 

93.0 

111.6 

130.2 

148.8 

167.4 

185 

18.5 

37.0 

55.5 

74.0 

92.5 

111.0 

129.5 

148.0 

166.5 

184 

18.4 

36.8 

55.2 

73.6 

92.0 

110.4 

128.8 

147.2 

165.6 

183 

18.3 

36.6 

54.9 

73.2 

91.5 

109.8 

128.1 

146.4 

164.7 

182 

18.2 

36.4 

54.6 

72.8 

91.0 

109.2 

127.4 

145.6 

163.8 

181 

18.1 

36.2 

54.3 

72.4 

90.5 

108.6 

126.7 

144.8 

162.9 

180 

18.0 

36.0 

54.0 

72.0 

90.0 

108.0 

126.0 

1440 

162.0 

179 

17.9 

35.8 

53.7 

71.6 

89.5 

107.4 

125.3 

143.2 

161.1 

146 


LOGARITHMS   OF  NUMBERS. 


No.  240  L.  380.J 


[No.  269  L.  431, 


N. 

1 

3 
4 
5 

6 

7 
8 
9 

250 

1 

2 
3 

5 
6 

7 

8 
9 

260 

2 
3 

4 
6 

8 
9 

0 

1 

2 

~0573 
2377 
4174 
5964 
7746 
9520 

3 

4 

5 

6 

7 

8 

T656 
3456 
5249 
7034 
8811 

9 

Diff. 

IsT 

180 
179 
178 
178 

177 
176 
176 
175 
174 

173 

173 
172 
171 

17! 
170 
169 

169 
168 
167 

167 
166 
165 

165 
164 
164 
163 
162 
162 

161 

380211 
2017 
3815 
5606 
7390 
9166 

0392 
2197 
3995 
5785 
7568 
9343 

0754 
2557 
4353 
6142 
7924 
9698 

0934 
2737 
4533 
6321 
8101 
9875 

1115 
2917 
4712 
6499 
8279 

1296 
3097 
4891 
6677 
8456 

1476 
3277 
5070 
6856 
8634 

1837 
3636 
5428 
7212 
8989 

0051 
1817 
3575 
5326 
7071 

8808 

0228 
1993 
3751 
5501 
7245 

8981 

0405 
2169 
3926 
5676 
7419 

9154 

0582 
2345 
4101 
5850 
7592 

9328 

0759 
2521 
4277 
6025 
7766 

9501 

390935 
2697 
4452 
6199 

7940 
9674 

1112 
2873 
4627 
6374 

8114 
9847 

1288 
3048 
4802 
6548 

8287 

1464 
3224 
4977 
6722 

8461 

1641 
3400 
5152 
6896 

8634 

0020 
1745 
3464 
'5176 
6881 
8579 

0192 
1917 
3635 
5346 
7051 
8749 

0365 
2089 
3807 
5517 
7221 
8918 

0538 
2261 
3978 
5688 
7391 
9087 

0711 
2433 
4149 
5858 
7561 
9257 

0883 
2605 
4320 
6029 
7731 
9426 

1056 
2777 
4492 
6199 
7901 
9595 

1228 
2949 
4663 
6370 
8070 
9764 

401401 
3121 
4834 
6540 
8240 
9933 

1573 
3292 
5005 
6710 
8410 

0102 
1788 
3467 

5140 
6807 
8467 

0271 
1956 
3635 

5307 
6973 
8633 

0440 
2124 
3803 

5474 
7139 
8798 

0609 
2293 
3970 

5641 
7306 
8964 

0777 
2461 
4137 

5808 
7472 
9129 

0946 
2629 
4305 

5974 
7638 
9295 

11  14 
2796 
4472 

6141 
7804 
9460 

1110 
2754 
4392 
6023 
7648 
9268 

1283 
2964 
4639 

6308 
7970 
9625 

1451 
3132 
4806 

6474 
8135 
9791 

1439 
3082 
4718 
6349 
7973 
9591 

411620 
3300 

4973 
6641 
8301 
9956 

0121 
1768 
3410 
5045 
6674 
8297 
9914 

0286 
1933 
3574 
5208 
6836 
8459 

0451 
2097 
3737 
5371 
6999 
8621 

0616 
2261 
3901 
5534 
7161 
8783 

0781 
2426 
4065 
5697 
7324 
8944 

0945 
2590 
4228 
5860 
7486 
9106 

1275 
2918 
4555 
6186 
7811 
9429 

421604 
3246 
4882 
6511 
8135 
9752 
43 

0075 

0236 

0398 

0559 

0720 

0881 

1042 

1203 

PROPORTIONAL  PARTS. 


Diff. 
T78~ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

17.8 

35.6 

53.4 

71.2 

89.0 

106.8 

124.6 

142.4 

160.2 

177 

17.7 

35.4 

53.1 

70.8 

88.5 

106.2 

•  23.9 

141.6 

159.3 

176 

17.6 

35.2 

52.8 

70.4 

88.0 

105.6 

123.2 

140.8 

158.4 

175 

17.5 

35.0 

52.5 

70.0 

87.5 

105.0 

122.5 

140.0 

157.5 

174 

17.4 

34.8 

52.2 

69.6 

87.0 

104.4 

121.8 

139.2 

156.6 

173 

173 

34.6 

51.9 

69.2 

86.5 

103.8. 

121.1 

138.4 

155.7 

172 

17.2 

34.4 

51.6 

68.8 

86.0 

103.2 

120.4 

137.6 

154.8 

171 

17.1 

34.2 

51.3 

68.4 

85.5 

102.6 

119.7 

136.8 

153.V 

170 

17.0 

34.0 

51.0 

68.0 

85.0 

102.0 

119.0 

136.0 

153.0 

169 

16.9 

33.8 

50.7 

67.6 

84.5 

101.4 

118.3 

135.2 

152.1 

168 

16.8 

33.6 

50.4 

67.2 

84.0 

100.8 

117.6 

134.4 

151.2 

167 

16.7 

33.4 

50.1 

66.8 

83.5 

100.2 

116.9 

133.6 

150.3 

166 

16.6 

33.2 

49.8 

66.4 

830 

99.6 

116.2 

132.8 

149.4 

165 

16.5 

33.0 

49.5 

66.0 

82.5 

99.0 

115.5 

132.0 

148.5 

164 

16.4 

32.8 

49.2 

65.6 

82.0 

98.4 

114.8 

131.2 

147.6 

163 

16.3 

32.6 

48.9 

65.2 

81.5           97.8 

114.1 

130.4 

146.7 

162 

16.2 

32.4 

48.5 

64.8 

81.0           97.2 

113.4 

129.6 

1458 

161 

16.1 

32.2 

48.3 

64.4 

80.5  1        96.6 

112.7 

128.8 

144.9 

LOGARITHMS   OF  NUMBERS.  147 

No.  270  L.  431.]  [No.  299  L.  476. 


N. 

?70 
1 
2 
3 
4 
5 

6 

8 
9 

280 

2 

4 
5 
6 
7 
8 

9 

290 

1 

3 

4 
5 

6 

8 
9 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

431364 
2969 
4569 
6163 
7751 
9333 

1525 
3130 
4729 
6322 
7909 
9491 

1685 
3290 
4888 
6481 
8067 
9648 

1846 
3450 
5048 
6640 
8226 
9806 

2007 
3610 
5207 
6799 
8384 
9964 

2167 
3770 
5367 
6957 
8542 

2328 
3930 
5526 
7116 
8701 

24tte> 
4090 
5685 

7275 
8859 

2649 
4249 
5844 
7433 
9017 

2809 
4409 
6004 
7592 
9175 

161 
160 
159 
159 
158 

158 
157 
157 
156 
155 

155 

154 
154 
153 
153 
152 
152 
15} 

151 
150 

150 
149 
149 
148 
148 

147 
146 
146 
146 
145 

0122 
1695 
3263 
4825 
6382 

7933 
9478 

0279 
1852 
3419 
4981 
6537 

8088 
9633 

0437 
2009 
3576 
5137 
6692 

8242 
9787 

0594 
2166 
3732 
5293 
6848 

8397 
9941 

0752 
2323 
3889 
5449 
7003 

8552 

440909 
2480 
4045 
5604 

7158 
8706 

1066 
2637 
4201 
5760 

7313 
8861 

1224 
2793 
4357 
5915 

7468 
QQ15 

1381 
2950 
4513 
6071 

7623 
9170 

1538 
3106 
4669 
6226 

7778 
9324 

0095 
1633 
3165 
4692 
6214 
7731 
9242 

450249 
1786 
3318 
4845 
6366 
7882 
9392 

0403 
1940 
3471 
4997 
6518 
8033 
9543 

0557 
2093 
3624 
5150 
6670 
8184 
9694 

0711 
2247 
3777 
5302 
6821 
8336 
9845 

0865 
2400 
3930 
5454 
6973 
8487 
9995 

1018 
2553 
4082 
5606 
7125 
8638 

1172 
2706 
4235 
5758 
7276 
8789 

1326 
2859 
4387 
5910 
7428 
8940 

1479 
3012 
4540 
6062 
7579 
9091 

0146 
1649 

3146 
4639 
6126 
7608 
9085 

0296 
1799 

3296 
4788 
6274 
7756 
9233 

0447 
1948 

3445 
4936 
6423 
7904 
9380 

0597 
2098 

3594 
5085 
6571 
8052 
9527 

0748 
2248 

3744 
5234 
6719 
8200 
•9675 

460898 

2398 
3893 
5383 
6868 
8347 
9322 

1048 

2548 
4042 
5532 
7016 
8495 
9969 

1198 

2697 
4191 
5680 
7164 
8643 

1348 

2847 
4340 
5829 
7312 
8790 

1499 

2997 
4490 
5977 
7460 
8938 

0116 
1585 
3049 
4508 
5962 

0263 
1732 
3195 
4653 
6107 

0410 
1878 
3341 
4799 
6252 

0557 
2025 
3487 
4944 
6397 

0704 
2171 
3633 
5090 
6542 

0851 
2318 
3779 
5235 
6687 

0998 
2464 
3925 
5381 
6832 

1145 
2610 
4071 
5526 
6976 

471292 
2756 
4216 
5671 

1438 
2903 
4362 
5816 

PROPORTIONAL,  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7. 

8 

9 

161 
160 

16.1 
16.0 

32.2 
32.0 

48.3' 
48.0 

64.4 
64.0 

80.5 
80.0 

96.6 
96.0 

112.7 
112.0 

128.8 
128.0 

144.9 
144.0 

159 

15.9 

31.8 

47.7 

63.6 

79.5 

95.4 

111.3 

127.2 

143.1 

158 

15.8 

31.6 

47.4 

63.2 

79.0 

94.8 

110.6 

126.4 

142.2 

157 

15.7 

31.4 

47.1 

62.8 

78.5 

94.2 

109.9 

125.6 

141.3 

156 

15.6 

31.2 

46.8 

62.4 

78.0 

93.6 

109.2 

124.8 

140.4 

155 

15.5 

31.0 

46.5 

62.0 

77.5 

93.0 

108  5 

124.0 

139.5 

154 

15.4 

30.8 

46.2 

61.6 

77.0 

92.4 

107.8 

123.2 

138.6 

153 

15.3 

30.6 

45.9 

61.2 

76:5 

91.8 

107.1 

122.4 

137.7 

152 

15.2 

30.4 

45.6 

60.8 

76.0 

91.2 

106.4 

121.6 

136.8 

151 

15.1 

30.2 

45.3 

60.4 

75.5' 

90.6 

105.7 

120.8 

135.9 

150 

15.0 

30.0 

450 

60.0 

75.0 

90.0 

105.0 

120.0 

135.0 

149 

14.9 

29.8 

44.7 

59.6 

74.5 

89.4 

104.3 

119.2 

134.1 

148 

14.8 

29.6 

44.4 

59.2 

74.0 

88.8 

103.6 

118.4 

133.2 

147 

14.7 

29.4 

44.1 

58.8 

73.5 

88.2 

102.9 

117.6 

132.3 

146 

14.6 

29.2 

43.8 

58.4 

73.0 

87.6 

102.2 

116.8 

131.4 

145 

14.5 

29.0 

43.5 

58.0 

72.5 

87.0 

101.5 

1160 

1305 

144 

14.4 

28.8 

43.2 

57.6 

72.0 

86.4 

100.8 

115.2 

129.6 

143 

14.3 

28.6 

42.9 

57.2 

71.5 

85.8 

100.1 

114.4 

128.7 

142 

14.2 

28.4 

42.6 

56.8 

71.0 

85.2 

99.4 

113.6 

127.8 

141 

14.1 

28.2 

42.3 

56.4 

70.5 

84.6 

98.7 

112.8 

126.9 

140 

14.0 

28.0 

42.0 

56.0 

70.0 

84.0 

98.0 

112.0  ! 

126.0 

LOGARITHMS   OF  NUMBERS. 


No.  300  L.  477.] 


[No.  339  L.  531. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

300 

1 

477121 
8566 

7266 
8711 

7411 
8855 

7555 
8999 

7700 
9143 

7844 
9287 

7989 
9431 

8133 
9575 

8278 
9719 

8422 
9863 

1  145' 
144 

2 
3 
4 
5 
6 
7 
8 

480007 
1443 
2874 
4300 
5721 
7138 
8551 

0151 
1586 
3016 
4442 
5863 
7280 
8692 

0294 
1729 
3159 
4585 
6005 
7421 
8833 

0438 
1372 
3302 
4727 
6147 
7563 
8974 

0582 
2016 
3445 
4869 
6289 
7704 
91  14 

0725 
2159 
3587 
5011 
6430 
7845 
9255 

0369 
2302 
3730 
5153 
6572 
7986 
9396 

1012 
2445 
3872 
5295 
6714 
8127 
9537 

1156 
2588 
4015 
5437 
6355 
8269 
9677 

1299 
2731 
4157 
5579 
6997 
8410 
9818 

144 
143 
143 
142 
142 
141 
141 

9 

9958 

0099 

0239 

0380 

0520 

0661 

0801 

0941 

1081 

1222 

140 

310 
1 
2 
3 
4 
5 

491362 
2760 
4155 
5544 
6930 
8311 

1502 
2900 
4294 
5683 
7068 
8448 

1642 
3040 
4433 
5822 
7206 
8586 

1782 
3179 
4572 
5960 
7344 
8724 

1922 
3319 
4711 
6099 
7483 
8862 

2062 
3458 
4850 
6238 
7621 
8999 

2201 
3597 
4989 
6376 
7759 
9137 

2341 
3737 
5128 
6515 
7897 
9275 

2481 
3876 
5267 
6653 
8035 
9412 

2621 
4015 
5406 
6791 
8173 
9550 

140 
139 
139 
139 
138 
138 

6 

9687 

9824 

9962 

0099 

0236 

0374 

0511 

0648 

0785 

0922 

137 

7 
8 
9 

320 
1 
2 

501059 
2427 
3791 

5150 
6505 
7856 

1  196 
2564 
3927 

5286 
6640 
7991 

1333 
2700 
4063 

5421 
6776 
8126 

1470 

2837 
4199 

5557 
6911 
8260 

1607 
2973 
4335 

5693 
7046 
8395 

1744 
3109 
4471 

5828 
7181 
8530 

1880 
3246 
4607 

5964 
7316 
8664 

2017 
3382 
4743 

6099 

7451 
8799 

2154 
3518 
4878 

6234 
7586 
8934 

2291 
3655 
5014 

6370 
7721 
9068 

137 
136 
136 

136 
135 
135 

3 

9203 

9337 

947  1 

9606 

9740 

9874 

0009 

0143 

0277 

041  1 

134 

4 
5 
6 
7 
8 
9 

330 

510545 
1883 
3218 
4548 
5874 
7196 

8514 

0679 
2017 
3351 
4681 
6006 
7328 

8646 

0813 
2151 
3484 
4813 
6139 
7460 

8777 

0947 
2284 
3617 
4946 
6271 
7592 

8909 

1081 
2418 
3750 
5079 
6403 
7724 

9040 

1215 
2551 
3883 
5211 
6535 
7855 

9171 

1349 
2684 
4016 
5344 
6668 
7987 

9303 

1482 
2818 
4149 
5476 
6800 
8119 

9434 

1616 
2951 
4282 
5609 
6932 
8251 

9566 

1750 
3084 
4415 
5741 
7064 
8382 

9697 

134 
133 
133 
133 
132 
132 

131 

1 

9828 

9959 

0090 

0221 

0353 

0484 

0615 

0745 

0876 

1007 

131 

2 
3 
4 
5 
6 
7 

521138 
2444 
3746 
5045 
6339 
7630 

1269 
2575 
3876 
5174 
6469 
7759 

1400 
2705 
4006 
5304 
6598 
7888 

1530 
2835 
4136 
5434 
6727 
8016 

1661 
2966 
4266 
5563 
6856 
8145 

1792 
3096 
4396 
5693 
6985 
8274 

1922 
3226 
4526 
5822 
7114 
8402 

2053 
3356 
4656 
5951 
7243 
8531 

2183 
3486 
4785 
6081 
7372 
8660 

2314 
3616 
4915 
6210 
7501 
8788 

131 
130 
130 
129 
129 
129 

8 

8917 

9045 

9174 

9302 

9430 

9559 

9687 

9815 

9943 

0072 

128 

9 

530200 

0328 

0456 

0584 

0712 

0840 

0968 

1096 

1223 

1351 

128 

PROPORTIONAL  PARTS. 


Diff 

1 

3 

3 

4 

5 

6 

7 

,      8 

9 

139 

13.9 

27.8 

41.7 

55.6 

69.5 

83.4 

97.3 

11  1.2 

125.1 

138 

13.8 

27.6 

41.4 

55.2 

69.0 

82.8 

96.6 

110.4 

124.2 

137 

13.7 

27.4 

41.1 

54.8 

68.5 

82.2 

95.9 

109.6 

123.3 

136 

13.6 

27.2 

40.8 

54.4 

68.0 

81.6 

95.2 

108.8 

122.4 

135 

13.5 

27.0 

40.5 

54.0 

67.5 

81.0 

94.5 

108.0 

121.5 

134 

13.4 

26.8 

40.2 

53.6 

67.0 

80.4 

93.8 

107.2 

120.6 

133 

13.3 

26.6 

39.9 

53.2 

66.5 

79.8 

93.1 

106.4 

119.7 

132 

13.2 

26.4 

39.6 

52.8 

66.0 

79.2 

92.4 

105.6 

118.8 

131 

13.1 

26.2 

39.3 

52.4 

65.5 

78.6 

91.7 

104.8 

117.9 

130 

13.0 

26.0 

39.0 

52.0 

65.0 

78.0 

91.0 

1040 

117.0 

129 

12.9 

25.8 

38.7 

51.6 

64.5 

77.4 

90.3 

103.2 

M6.1 

128 

12.8 

25.6 

38.4 

51.2 

64  0 

76.8 

89.6 

102.4 

115.2 

127 

12.7 

25.4  * 

38.1 

50.8  1 

63.5 

76.2 

88.9     1 

101.6 

114.3 

LOGARITHMS    OF   NUMBERS. 


149 


No.  340  L.  531.J  ' 


[No. 379  L.  579. 


N. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

Diff. 

340 
2 

4 
5 
6 

7 
8 
9 

350 
1 

3 
4 

5 
6 

8 
9 

360 
1 
2 
3 

4 
5 

6 

8 
9 

370 

I 

2 

4 
5 
6 

8 
9 

531479 
2754 
4026 
5294 
6558 
7819 
9076 

1607 
2882 
4153 
5421 
6685 
7945 
9202 

1734 
3009 
4280 
5547 
6811 
8071 
9327 

1862 
3136 
4407 
5674 
6937 
8197 
9452 

1990 
3264 
4534 
5800 
7063 
8322 
9578 

2117 
3391 
4661 
5927 
7189 
8448 
9703 

2245 
3518 
4787 
6053 
7315 
8574 
9829 

2372 
3645 
4914 
6180 
7441 
8699 
9954 

2500 
3772 
5041 
6306 
7567 
8825 

2627 
3899 
5167 
6432 
7693 
8951 

128 
127 
127 
126 
126 
126 

125 
125 
125 
124 

124 
124 
123 
123 

123 
122 
122 
121 
121 
121 

120 
120 
120 

119 
119 
119 

m 

118 
118 
118 

117 

117 
117 
116 
116 
116 
115 
115 
115 
114 

0079 
1330 
2576 
3820 

5060 
6296 
7529 
8758 
9984 

0204 
1454 
2701 
3944 

5183 
6419 
7652 
8881 

540329 
1579 
2825 

4068 
5307 
6543 
7775 
9003 

0455 
1704 
2950 

4192 
5431 
6666 
7898 
9126 

0580 
1829 
3074 

4316 
5555 
6',  89 
8021 
9249 

0705 
1953 
3199 

4440 
5678 
6913 
8144 
9371 

0830 
2078 
3323 

4564 
5802 
7036 
8267 
9494 

0955 
2203 

3447 

4688 
5925 
7159 
8389 
9616 

1080 
2327 
3571 

4812 
6049 
7282 
8512 
9739 

1205 
2452 
3696 

4936 
6172 
7405 
8635 
9861 

0106 
1328 
2547 
3762 
4973 
6«82 

7387 
8589 
9787 

550228 
1450 
2668 
3883 
5094 

6303 
7507 
8709 
9907 

0351 
1572 
2790 
4004 
5215 

6423 
7627 
8829 

0473 
1694 
2911 
4126 
5336 

6544 
7748 
8948 

0595 
1816 
3033 

4247 
5457 

6664 
7868 
9068 

0717 
1938 
3155 
4368 
5578 

6785 
7988 
9188 

0840 
2060 
3276 
4489 
5699 

6905 
8108 
9308 

0962 
2181 
3398 
4610 
5820 

7026 
8228 
9428 

1084 
2303 
3519 
4731 
5940 

7146 
8349 
9548 

1206 
2425 
3640 
4852 
6061 

7267 
8469 
9667 

0026 
1221 
2412 
3600 
4784 
5966 
7144 

8319 
9491 

0146 
1340 
2531 
3718 
4903 
6084 
7262 

8436 
9608 

0265 
1459 
2650 
3837 
5021 
6202 
7379 

8554 
9725 

0385 
1578 
2769 
3955 
5139 
6320 
7497 

8671 
9842 

0504 
1698 
2887 
4074 
5257 
6437 
7614 

8788 
9959 

0624 
1817 
3006 
4192 
5376 
6555 
7732 

8905 

0743 
1936 
3125 
4311 
5494 
6673 
7849 

9023 

0863 
2055 
3244 
4429 
5612 
6791 
7967 

9140 

0982 

2174 
3362 
4548 
5730 
6909 
8084 

9257 

561101 
2293 
3481 
4666 
5848 
7026 

8202 
9374 

0076 
1243 
2407 
3568 
4726 
5880 
7032 
8181 
9326 

0193 
1359 
2523 
3684 
4841 
5996 
7147 
8295 
9441 

0309 
1476 
2639 
3800 
4957 
6111 
7262 
8410 
9555 

0426 
1592 
2755 
3915 
5072 
6226 
73,77 
8525 
9669 

570543 
1709 
2872 
4031 
5188 
6341 
7492 
8639 

0660 
1825 
2988 
4147 
5303 
6457 
7607 
8754 

0776 
1942 
3104 
4263 
5419 
6572 
7722 
8868 

0893 
2058 
3220 
4379 
5534 
6687 
7836 
8983 

1010 
2174 
3336 
4494 
5650 
6802 
7951 
9097 

1126 
2291 
3452 
4610 
5765 
6917 
8066 
9212 

PROPORTIONAL  PARTS. 


Diff. 

1 

3 

3 

4 

5 

6 

7 

8 

9 

128 

12.8 

25.6 

38.4 

51.2 

64.0 

76.8 

89.6 

102.4 

115.2 

127 

12.7 

25.4 

38.1 

50.8 

63.5 

76.2 

88:9 

101.6 

114.3 

126 

12.6 

25.2 

37.8 

50.4 

63.0 

75.6 

88.2 

100.8 

113.4 

125 

12.5 

25.0 

37.5 

50.0 

62.5 

75.0 

87.5 

100.0 

112.5 

124 

12.4 

24.8 

37.2 

49.6 

62.0 

74.4 

86.8 

99.2 

111.6 

123 

12.3 

24.6 

36.9 

49.2 

61.5 

73.8 

86.1 

98.4 

110.7 

122 

12.2 

24.4 

36.6 

48.8 

61.0 

73.2 

85.4 

97.6 

109.8 

121 

12.1 

24.2 

36.3 

48.4 

60.5 

72.6 

84.7 

96.8 

108.9 

120 

12.0 

24.0 

36.0 

48.0 

60.0 

72.0 

84.0 

96.0 

108.0 

119 

11.9 

23.8 

35.7 

47.6 

59.5 

71.4 

83.3 

95.2 

107.1 

150 


LOGARITHMS    OF   NUMBERS. 


No.  380  L.  579.J 


[No.  414  L.  617. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 
114 

113 
112 

111 

110 
109 

108 
107 

106 
105 

380 

1 
2 

4 

6 

7 
8 
9 

390 
1 
2 
3 
4 
5 
6 
7 
8 

9 

400 
1 
2 
3 
4 
5 
6 
7 

8 
9 

410 
2 
4 

579784 

9898 

0012 
1153 
2291 
3426 
4557 
5686 
6812 
7935 
9056 

0126 
1267 
2404 
3539 
4670 
5799 
6925 
8047 
9167 

0241 
1381 
2518 
3652 
4783 
5912 
7037 
8160 
9279 

0355 
1495 
2631 
3765 
4896 
6024 
7149 
8272 
9391 

0469 
1608 
2745 
3879 
5009 
6137 
7262 
8384 
9503 

0583 
1722 
2858 
3992 
5122 
6250 
7374 
8496 
9615 

0697 
1836 
2972 
4105 
5235 
6362 
7486 
8608 
9726 

0811 
1950 
3085 
4218 
5348 
6475 
7599 
8720 
9838 

580925 
2063 
3199 
4331 
5461 
6587 
7711 
8832 
9950 

1039 
2177 
3312 
4444 
5574 
6700 
7823 
8944 

0061 

1176 
2288 
3397 
4503 
5606 
6707 
7805 
8900 
9992 

~\OS2 

2169 
3253 
4334 
5413 
6489 
7562 
8633 
9701 

0173 

1287 
2399 
3508 
4614 
5717 
6817 
7914 
9009 

0284 

1399 
2510 

3618 
4724 
5827 
6927 
8024 
9119 

0396 

1510 
2621 
3729 
4834 
5937 
7037 
8134 
9228 

0507 

1621 
2732 
3840 
4945 
6047 
7146 
8243 
9337 

0619 

1732 
2843 
3950 
5055 
6157 
7256 
8353 
9446 

0730 

1843 
2954 
4061 
5165 
6267 
7366 
8462 
9556 

0842 

1955 
3064 
4171 
5276 
6377 
7476 
8572 
9665 

0953 

2066 
3175 
4282 
5386 
6487 
7586 
8681 
9774 

591065 
2177 
3286 
4393 
5496 
6597 
7695 
8791 
9883 

0101 
1191 

2277 
3361 
4442 
5521 
6596 
7669 
8740 
9808 

0210 
1299 

2386 
3469 
4550 
5628 
6704 
7777 
8847 
9914 

0319 
1406 

2494 
3577 
4658 
5736 
681  1 
7884 
8954 

0428 
1517 

2603 
3686 
4766 
5844 
6919 
7991 
9061 

0537 
1625 

2711 
3794 
4874 
5951 
7026 
8098 
9167 

0646 
1734 

2819 
3902 
4982 
6059 
7133 
8205 
9274 

0755 
1843 

2928 
4010 
5089 
6166 
7241 
8312 
9381 

0864 
1951 

3036 
4118 
5197 
6274 
7348 
8419 
9488 

600973 

2060 
3144 
4226 
5305 
6381 
7455 
8526 
9594 

0021 
1086 
2148 

3207 
4264 
5319 
6370 
7420 

0128 
1192 
2254 

3313 
4370 
5424 
6476 
7525 

0234 
1298 
2360 

3419 
4475 
5529 
6581 
7629 

0341 
1405 
2466 

3525 
4581 
5634 
6686 
7734 

0447 
1511 
2572 

3630 
4686 
5740 
6790 
7839 

0554 
1617 
2678 

3736 
4792 
5845 
6895 
7943 

610660 
1723 

2784 
3842 
4897 
5950 
7000 

0767 
1829 

2890 
3947 
5003 
6055 
7105 

0873 
1936 

2996 
4053 
5108 
6160 
7210 

0979 
2042 

3102 
4159 
5213 
6265 
7315 

PROPORTIONAL  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

118 

11.8 

23.6 

35.4 

47.2 

59.0 

70.8 

82.6 

94.4 

106.2 

117 

11.7 

23.4 

35.1 

46.8 

58.5 

70.2 

81.9 

93.6 

105.3 

116 

11.6 

23.2 

34.8 

46.4 

58.0 

69.6 

81.2 

92.8 

104.4 

115 

11.5 

23.0 

34.5 

46.0 

57.5 

69.0 

80.5 

92.0 

103.5 

114 

11.4 

22.8 

34.2 

45.6 

57.0 

68.4 

79.8 

9,  7 

102.6 

113 

11.3 

22.6 

33.9 

45.2 

56.5 

67.8 

79.1 

90.4 

101.7 

112 

11.2 

22.4 

33.6 

44.8 

56.0 

67.2 

78.4 

89.6 

100.8 

111 

11.1 

22.2 

33.3 

44.4 

55.5 

66.6 

77.7 

88.8 

99.9 

110 

11.0 

22.0 

33.0 

44.0 

55.0 

66.0 

77.0 

88.0 

99.0 

109 

10.9 

21.8 

32.7 

43.6 

54.5 

65.4 

76.3 

87.2 

98.1 

108 

10.8 

21.6 

32.4 

43.2 

54.0 

64.8 

75.6 

86.4 

97.2 

107 

10.7 

21.4 

32.1 

42.8 

53.5 

64.2 

74.9 

85.6 

96.3 

106 

10.6 

21.2 

31.8 

42.4 

53.0 

63.6 

74.2 

84.8 

95.4 

105 

10.5 

21.0 

31.5 

42.0 

52.5 

63.0 

73.5 

84.0 

94.5 

104 

10.4 

20.8 

31.2 

41.6 

52.0 

62.4 

72.8 

83.2 

93.6 

LOGARITHMS    OF    NUMBERS. 


151 


No.  415  L.  618.] 


[No.  459  L.  662. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

415 

618048 

8153 

8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

~105 

6 

9093 

9198 

9302 

9406 

9511 

9615 

9719 

9824 

9928 

"0032 

7 

620136 

0240 

0344 

0448 

0552 

0656 

0760 

0864 

0968 

1072 

104 

8 

1176 

1280 

1384 

1488 

1592 

1695 

1799 

1903 

2007 

2110 

9 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3146 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

1 

4282 

4385 

4488 

4591 

4695 

4793 

4901 

5004 

5107 

5210 

103 

2 

5312 

5415 

5518 

5621 

5724 

5827 

5929 

6032 

6135 

6238 

3 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7058 

7161 

7263 

4 

7366 

7463 

7571 

7673 

7775 

7878 

7980 

8082 

8185 

8287 

5 

8389 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

.102 

6 

9410 

9512 

9613 

9715 

9817 

9919 

0021 

0123 

0224 

0326 

7 

630428 

0530 

~063T 

0733 

0835 

0936 

1038 

1139 

1241 

1342 

8 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

2153 

2255 

2356 

9 

2457 

2559 

2660 

2761 

2862 

2963 

3064 

3165 

3266 

3367 

430 

3468 

3569 

3670 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

101 

1 

4477 

4578 

4679 

4779 

4880 

4981 

5081 

5182 

5283 

5383 

2 

5484 

5584 

5635 

5785 

5886 

5986 

6087 

6187 

6287 

6388 

3 

6488 

6588 

6638 

6789 

6889 

6989 

7089 

7189 

7290 

7390 

4 

7490 

7590 

7690 

7790 

7890 

7990 

8090 

8190 

8290 

8389 

100 

5 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

9287 

9387 

6 

9486 

9586 

9686 

9785 

9885 

9984 

0084 

0183 

0283 

0382 

7 

640431 

0581 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

8 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

9 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

3354 

99 

440 

3453 

3551 

3650 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

1 

4439 

4537 

4636 

4734 

4832 

4931 

5029 

5127 

5226 

5324 

2 

5422 

5521 

5619 

5717 

5815 

5913 

6011 

6110 

6208 

6306 

3 

6404 

6502 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

98 

4 

7333 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165 

8262 

5 

8360 

8458 

8555 

8653 

8750 

8848 

8945 

9043 

9140 

9237 

6 

9335 

9432 

9530 

9627 

9724 

9821 

9919 

0016 

0113 

0210 

7 

650303 

0405 

0502 

"0599 

0696 

0793 

0890 

0987 

1084 

1181 

8 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

2150 

97 

9 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019 

3116 

450 

3213 

3309 

3405 

3502 

3598 

3695 

3791 

3888 

3984 

4080 

4177 

4273 

4369 

4465 

4562 

4658 

4754 

4850 

4946 

5042 

2 

5138 

5235 

5331 

5427 

5523 

5619 

5715 

5810 

5906 

6002 

96 

3 

6098 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

4 

7056 

7152 

7247 

7343 

7438 

7534 

7629 

7725 

7820 

7916 

5 

8011 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

8870 

6 

8965 

9060 

9155 

9250 

9346 

9441 

9536 

9631 

9726 

9821 

7 

9916 

0011 

0106 

0201 

0296 

0391 

0486 

0581 

0676 

"0771 

95 

8 

660865 

0960 

1055 

1150 

1245 

1339 

1434 

1529 

1623 

1718 

9 

1813 

1907 

2002 

2096 

2191 

2286 

23801  2475 

2569 

2663 

PROPORTIONAL  PARTS. 


Diff. 

1 

10.5 
10.4 
10.3 
10.2 
10.1 
10.0 
9.9 

2 

3 

4 

5 

6 

7 

8 

9 

105 
104 
103 
102 
101 
100 
99 

21.0 
20.8 
20.6 
20.4 
20.2 
20.0 
19.8 

31.5 
31.2 
30.9 
30.6 
30.3 
30.0 
29.7 

42.0 
41.6 
41.2 
40.8 
40.4 
40.0 
39.6 

52.5 
52.0 
51.5 
51.0 
50.5 
50.0 
49.5 

63.0 
62.4 
61.8 
61.2 
60.6 
60.0 
59.4 

73.5 
72.8 
72.1 
71.4 
70.7 
70.0 
69.3 

84.0 
83.2 
82.4 
81.6 
80.8 
80.0 
79.2 

94.5 
93.6 
92.7 
91.8 
90.9 
90.0 
89.1 

152 


LOGARITHMS   OF  NUMBERS. 


No.  460  L.  662.] 


[No.  499  L.  698 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff< 

"46CT 

'  662758 

2852 

2947 

3041 

3135 

3230 

"3324 

3418 

3512 

3607 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

4360 

4454 

4548 

2 

4642 

4736 

4830 

4924 

5018 

5112 

5206 

5299 

5393 

5487 

94 

3 

5581 

5675 

5769 

5862 

5956 

6050 

6143 

6237 

6331 

6424 

4 

•6518 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

5 

7453 

7546 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

6 

8386 

8479 

8572 

8665 

8759 

8852 

8945 

9038 

9131 

9224 

7 

9317 

94  1C 

9503 

9596 

9689 

9782 

9875 

9967 

0060 

fil  53 

Ql 

8 

670246 

0339 

0431 

0524 

0617 

0710 

0802 

0895 

0988 

U  1  Jj 

1080 

7J 

9 

1173 

1265 

1358 

1451 

1543 

1636 

1728 

1821 

1913 

2005 

470 

2098 

2190 

2283 

2375 

2467 

2560 

2652 

2744 

2836 

2929 

1 

3021 

3rl13 

3205 

3297 

3390 

3482 

3574 

3666 

3758 

3850 

2 

3942 

4934 

4126 

4218 

4310 

4402 

4494 

4586 

4677 

4769 

92 

3 

4861 

4953 

5045 

5137 

5228 

5320 

5412 

5503 

5595 

5687 

4 

5778 

5870 

5962 

6053 

6145 

6236 

6328 

6419 

6511 

6602 

5 

6694 

6785 

6876 

6968 

7059 

7151 

7242 

7333 

7424 

7516 

6 

7607 

7698 

7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

7 

8518 

8609 

8700 

8791 

8882 

8973 

9064 

9155 

9246 

9337 

9! 

3 

9428 

9519 

9610 

9700 

9791 

9882 

9973 

0063 

01  54 

0745 

9 

680336 

0426 

0517 

0607 

0698 

0789 

0879 

0970 

1060 

\)mj 

1151 

480 

1241 

1332 

1422 

1513 

1603 

1693 

1784 

1874 

1964 

2055 

1 

2145 

2235 

2326 

2416 

2506 

2596 

2686 

2777 

2867 

2957 

2 

3047 

3137 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

3 

3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

4 

4845 

4935 

5025 

5114 

5204 

5294 

5383 

5473 

5563 

5652 

5 

5742 

5831 

5921 

601-0 

6100 

6189 

6279 

6368 

6458 

6547 

6 

6636 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

7 

7529 

7618 

7707 

77% 

7886 

7975 

8064 

8153 

8242 

8331 

8 

8420 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9131 

9220 

89 

9 

9309 

9398 

9486 

9575 

9664 

9753 

9841 

9930 

0019 

01(17 

490 

690196 

0285 

0373 

0462 

0550 

0639 

0728 

0816 

0905 

U  1  U/ 

0993 

1 

1081 

1170 

1258 

1347 

1435 

1524 

1612 

1700 

1789 

1877 

2 

1965 

2053 

2142 

2230 

2318 

2406 

2494 

2583 

2671 

2759 

3 

2847 

2935 

3023 

3111 

3199 

3287 

3375 

3463 

3551 

3639 

88 

4 

3727 

3815 

3903 

3991 

4078 

4166 

4254 

4342 

4430 

4517 

5 

4605 

4693 

4781 

4868 

4956 

5044 

5131 

5219 

5307 

5394 

6 

5482 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

7 

6356 

6444 

6531 

6618 

6706 

6793 

6880 

6968 

7055 

7142 

8 

7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

87 

9 

8100 

8188 

8275 

8362 

8449 

8535 

8622 

8709 

8796 

8883 

PROPORTIONAL  PARTS. 


Diff. 

1 

3 

3 

4 

5 

6 

7 

8 

9 

^98 

9.8 

19.6 

29  A 

39.2 

49.0 

58.8 

68.6 

78.4 

88.2 

97 

9.7 

19.4 

29.1 

38.8 

48.5 

58.2 

67.9 

77.6 

87.3 

96 

9.6 

19.2 

28.8 

38.4 

48.0 

57.6 

67.2 

76.8 

86.4 

95 

9.5 

19.0 

28.5 

38.0 

47.5 

57.0 

66.5 

76.0 

85.5 

94 

9.4 

18.8 

28.2 

37.6 

47.0 

56.4 

65.8 

75.2 

84.6 

93 

9.3 

18.6 

27.9 

37.2 

46.5 

55.8 

65.1 

74.4 

83.7 

92 

9.2 

18.4 

27.6 

36.8 

46.0 

55.2 

64.4 

73.6 

82.8 

91 

9.1 

18.2 

27.3 

36.4 

45.5 

54.6 

63.7 

72.8 

81.9 

90 

9.0 

18.0 

27.0 

36.0 

45.0 

54.0 

63.0 

72.0 

81.0 

89 

8.9 

17.8 

26.7 

35.6 

44.5 

53.4 

62.3 

71.2 

80.1 

88 

8.8 

17.6 

26.4 

35.2 

44.0 

52.8 

61.6 

70.4 

79.2 

87 

8.7 

17.4 

26.1 

34.8 

43.5 

52.2 

60.9 

69.6 

78.3 

66 

8.6 

17.2 

25.8 

34.4 

43.0 

51.6 

60.2 

68.8 

77.4 

LOGARITHMS   OF   NUMBERS. 


153 


No.  500  L.  698.1 


[No.  544  L.  736 


N. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

DiS. 

500 

698970 

QO-1Q 

9057 

QQ9.4 

9144 

9231 

9317 

9404 

9491 

"9578 

~9664 

9751 

t 

yoJO 

yy^^t 

001  1 

0098 

0184 

0271 

0358 

0444 

0531 

0617 

2 

700704 

0790 

0877 

0963 

1050 

1136 

1222 

1309 

1395 

1482 

3 

1568 

1654 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

4 

243  1 

2517 

2603 

2689 

2775 

2861 

2947 

3033 

3119 

3205 

5 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

3893 

3979 

4065 

86 

6 

4151 

4236 

4322 

4408 

4494 

4579 

4665 

4751 

4837 

4922 

7 

5008 

5094 

5179 

5265 

5350 

5436 

5522 

5607 

5693 

5778 

8 

5864 

5949 

6035 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

9 

6718 

6803 

6888 

6974 

7059 

7144 

7229 

7315 

7400 

7485 

510 

7570 

7655 

7740 

7826 

7911 

7996 

8081 

8166 

8251 

8336 

1 

8421 

8506 

8591 

8676 

8761 

8846 

8931 

9015 

9100 

9185 

85 

O?7fl 

Q-l  CC 

Q44O 

QCTX 

9609 

9694 

9779 

9863 

9948 

y^/U 

\TJ  JJ 

y^fnU 

•  y.?<6if 

0033 

3 

710117 

0202 

0287 

0371 

0456 

0540 

0625 

0710 

0794 

0879 

4 

0963 

1048 

1132 

1217 

1301 

1385 

1470 

1554 

1639 

1723 

5 

1807 

1892 

1976 

2060 

2144 

2229 

2313 

2397 

2481 

2566 

6 

2650 

2734 

2818 

2902 

2986 

3070 

3154 

3238 

3323 

3407 

7 

3491 

3575 

3659 

3742 

3826 

3910 

3994 

4078 

4162 

4246 

84 

8 

4330 

4414 

4497 

4581 

4665 

4749 

4833 

4916 

5000 

5084 

9 

5167 

5251 

5335 

5418 

5502 

5586 

5669 

5753 

5836 

5920 

520 

6003 

6087 

6170 

6*254 

6337 

6421 

6504 

6588 

6671 

6754 

6838 

6921 

7004 

7088 

7171 

7254 

7338 

7421 

7504 

7587 

2 

7671 

7754 

7837 

7920 

8003 

8086 

8169 

8253 

8336 

8419 

83 

3 

8502 

8585 

8668 

875T 

8834 

8917 

9000 

9083 

9165 

9248 

Qa-i  | 

QA1  / 

Q  AQ7 

QCOA 

Qf\f\^ 

Q7  AC* 

QQOQ 

991  1 

9994 

yjJ  1 

y^f  1  f 

y^ty/ 

VDOL 

yOO.7 

y/T-j 

yo^o 

0077 

5 

720159 

0242 

0325 

0407 

0490 

0573 

0655 

0738 

0821 

0903 

6 

0986 

1068 

1151 

1233 

1316 

1398 

1481 

1563 

1646 

1728 

7 

1811 

1893 

1975 

2058 

2140 

2222 

2305 

2387 

2469 

2552 

8 

2634 

2716 

2798 

2881 

2963 

3045 

3UZ7 

3209 

3291 

3374 

9 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

82 

530 

4276 

4358 

4440 

4522 

4604 

4685 

4767 

4849 

4931 

5013 

1 

5095 

5176 

5258 

5340 

5422 

5503 

5585 

5667 

5748 

5830 

5912 

5993 

6075 

6156 

6238 

6320 

6401 

6483 

6564 

6646 

3 

6727 

6809 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

7541 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

5 

8354 

8435 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

6 
7 

9165 
9974 

9246 

9327 

9408 

9489 

9570 

9651 

9732 

9813 

9893 

81 

005 

0136 

021  7 

029? 

0376 

045C 

0540 

0621 

0702 

8 

730782 

0863 

094^ 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

9 

1589 

1669 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

2313 

540 

2394 

2474 

2555 

2635 

2715 

2796 

2876 

2956 

3037 

3117 

3197 

3278 

3358 

3438 

3518 

3598 

3679 

3759 

3839 

3919 

2 

3999 

4079 

4160 

4240 

4320 

4400 

4480 

4560 

4640 

4720 

80 

3 

4800 

4880 

4960 

5040 

5120 

5200 

5279 

5359 

5439 

5519 

4 

5599 

5679 

5759 

5838 

5918 

5998 

6078 

6157 

6237 

6317 

PROPORTIONAL  PARTS. 


DIS. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

87 
86 
85 
84 

8.7 
8.6 
8.5 
8.4 

17.4 
17.2 
17.0 
16.8 

26.1 
25.8 
25.5 
25.2 

34.8 
34.4 
34.0 
33.6 

43.5 
43.0 
42.5 
42.0 

52.2 
51.6 
51.0 
50.4 

60.9 
60.2 
59.5 
58.8 

69.6 
68.8 
68.0 
67.2 

78.3 
77.4 
76.5 
75.6 

154 


LOGARITHMS   OF   NUMBERS. 


No.  545  L.  736.] 


[No  584  L.  767 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9  • 

Diff. 

$45 
6 

8 
9 

736397 
7193 
7987 
8781 
9572 

6476 
7272 
8067 
8860 
9651 

6556 
7352 
8146 
8939 
9731 

6635 
7431 
8225 
9018 
9810 

6715 

7511 
8305 
9097 
9889 

6795 
7590 
8384 
9177 
9968 

6874 
7670 
8463 
9256 

6954 
7749 
8543 
9335 

7034 
7829 
8622 
9414 

7113 
7908 
8701 
9493 

0047 

0126 

07OS 

nooj 

7O 

550 

740363 

0442 

0521 

0600 

0678 

0757 

Uvn/ 

0836 

0915 

U<iUJ 

0994 

v^OH 

1073 

/y 

1152 

1230 

1309 

1388 

1467 

1546 

1624 

1703 

1782 

1860 

2 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2647 

3 

2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

4 

3510 

3588 

3667 

3745 

3823 

3902 

3980 

4058 

4136 

4215 

5 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

6 

5075 

5153 

5231 

5309 

5387 

5465 

5543 

5621 

5699 

5777 

78 

7 

5855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

8 

6634 

6712 

6790 

6868 

6945 

7023 

7101 

7179 

7256 

7334 

9 

7412 

7489 

7567 

7645 

7722 

7800 

7878 

7955 

8033 

8110 

560 

8188 

8266 

8343 

8421 

8498 

8576 

8653 

8731 

8808 

8885 

8963 

9040 

9118 

9195 

9272 

9350 

9427 

9504 

9582 

9659 

2 

9736 

9814 

9891 

9968 

0045 

0123 

0200 

0277 

0354 

0431 

3 

750508 

0586 

0663 

0740 

0817 

0894 

0971 

1048 

1125 

1202 

4 

1279 

1356 

1433 

1510 

1587 

1664 

.  1741 

1818 

1895 

1972 

77 

5 

2048 

2125 

2202 

2279 

2356 

2433 

2509 

2586 

2663 

2740 

6 

2816 

2893 

2970 

3047 

3123 

3200 

3277 

3353 

3430 

3506 

7 

3583 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4195 

4272 

8 

4348 

4425 

4501 

4578 

4654 

4730 

4807 

4883 

4960 

5036 

9 

5112 

5189 

5265 

5341 

5417 

5494 

5570 

5646 

5722 

5799 

570 

5875 

5951 

6027 

6103 

6180 

6256 

6332 

6408 

6484 

6560 

1 

6636 

6712 

6758 

6864 

6940 

7016 

7092 

7168 

7244 

7320 

76 

2 

7396 

7472 

7548 

7624 

7700 

7775 

7851 

7927 

8003 

8079 

3 

8155 

8230 

8306 

8382 

8458 

8533 

8609 

8685 

8761 

8836 

4 

8912 

8988 

9063 

9139 

9214 

9290 

9366 

9441 

9517 

9592 

i 

QAA« 

074-1 

9819 

9894 

QQ7O 

9 

7ODO 

7/*1J 

77/\J 

0045 

0121 

0196 

0272 

0347 

6 

760422 

0498 

0573 

0649 

0724 

0799 

0875 

0950 

1025 

1101 

7 

1176 

1251 

1326 

1402 

1477 

1552 

1627 

1702 

1778 

1853 

8 

1928 

2003 

2078 

2153 

2228 

2303 

2378 

2453 

2529 

2604 

73 

9 

2679 

2754 

2829 

2904 

2978 

3053 

3128 

3203 

3278 

3353 

580 

3428 

3503 

3578 

3653 

3727 

3802 

3877 

3952 

4027 

4101 

1 

4176 

4251 

4326 

4400 

4475 

4550 

4624 

4699 

4774 

4848 

2 

4923 

4998 

5072 

5147 

5221 

5296 

5370 

5445 

5520 

5594 

3 

5669 

5743 

5818 

5892 

5966 

6041 

6115 

6190 

6264 

6338 

4 

6413 

6487 

6562 

6636 

6710 

6785 

6859 

6933 

7007 

7082 

PROPORTIONAL,  PARTS. 


Diff. 
1ST- 

1 

3 

~T6T 

3 

4 

5 

6 

7 

8 

9 

8.3 

24.9 

33.2 

41.5 

49.8 

58.1 

66.4 

74.7" 

82 

8.2 

16.4 

24.6 

32.8 

41.0 

49.2 

57.4 

65.6 

73.8 

81 

8.1 

16.2 

24.3 

32.4 

40.5 

48.6 

56.7 

64.8 

72.9 

80 

8.0 

16.0 

24.0 

32.0 

40.0 

48.0 

56.0 

64.0 

72.0 

79 

7.9 

15.8 

23.7 

31.6 

39.5 

47.4 

55.3 

63.2 

71.1 

78 

7.8 

15.6 

23.4 

31.2 

39.0 

46.8 

54.6 

62.4 

70.2 

77 

7.7 

15.4 

23.1 

30.8 

38.5 

46.2 

53.9 

61.6 

69.3 

76 

7.6 

15.2 

22.8 

30.4 

38.0 

45.6 

53.2 

60.8 

68.4 

75 

7.5 

15.0 

22.5 

30.0 

37.5 

45.0 

52.5 

60.0 

67.5 

74 

7.4 

14,8 

22,2 

29.6 

37.0 

44.4 

51.8 

59.3 

66* 

LOGARITHMS   OF   NUMBERS. 


155 


No.  585  L.  767.] 


[No.  629  L.  79ft 


N. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

Diff. 

585 

767156 

7230 

~730~4 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

6 

7898 

7972 

8046 

8120 

8194 

8268 

8342 

8416 

8490 

8564 

74 

7 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9156 

9230 

9303 

9377 

945  1 

9525 

9599 

9673 

9746 

9820 

9894 

9968 

0042 

9 

770115 

0189 

0263 

0336 

0410 

0484 

0557 

0631 

0705 

0778 

590 

0852 

0926 

0999 

1073 

1146 

1220 

1293 

1367 

1440 

1514 

1 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

2 

2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

3 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

4 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

5 

4517 

4590 

4663 

4736 

4809 

4882 

4955 

5028 

5100 

5173 

6 

5246 

5319 

5392 

5465 

5538 

5610 

5683 

5756 

5829 

5902 

7 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

8 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

9 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

600 

8151 

8224 

8296 

8368 

8441 

8513 

8585 

8658 

8730 

8802 

1 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

9596 

9669 

9741 

9813 

9885 

9957 

0029 

0101 

0173 

0245 

3 

780317 

0389 

0461 

0533 

0605 

0677 

0749 

0821 

0893 

0965 

72 

4 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

5 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

6 

2473 

2544 

2616 

268  T 

2759 

2831 

2902 

2974 

3046 

3117 

7 

3189 

3260 

3332 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

8 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

9 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

610 

5330 

5401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

5970 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

71 

2 

6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

7390 

7460 

7531 

7602 

7673 

7744 

7815 

7885 

7956 

8027 

8098 

4 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

5 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

9581 

965  1 

9722 

9792 

986- 

9933 

0004 

0074 

0144 

0215 

7 

790285 

0356 

0426 

0496 

0567 

0637 

0707 

0778 

0848 

0918 

8 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

9 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

620 

2392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

1 

3092 

3162 

3231 

3301 

3371 

3441 

3511 

3581 

3651 

3721 

2 

3790 

3860 

3930 

4000 

4070 

4139 

4209 

4279 

4349 

4418 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

5115 

4 

5185 

5254 

5324 

5393 

5463 

5532 

5602 

5672 

5741 

5811 

5 

5880 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

6 

6574 

6644 

6713 

6782 

6852 

6921 

6990 

7060 

7129 

7198 

7 

7268 

7337 

7406 

7475 

7545 

7614 

7683 

7752 

7821 

7890 

8 

7960 

8029 

8098 

8167 

8236 

8305 

8374 

8443 

8513 

8582 

9 

8651 

8720 

8789 

8858 

8927 

8996 

9065 

9134  9203 

9272 

69 

PROPORTIONAL  PARTS. 


Diff. 

74 
73 
72 
71 
70 
69 

1 

2 

3 

4 

5 

6 

7 

8 

9 

7.5 

7.4 
7.3 
7.2 
7.1 
7.0 
6.9 

15.0 

14.8 
14.6 
14.4 
14.2 
14.0 
13.8 

22.5 
22.2 
21.9 
21.6 
21.3 
21.0 
20.7 

30.0 
29.6 
29.2 
28.8 
28.4 
28.0 
27.6 

37.5 
37.0 
36.5 
36.0 
35.5 
35.0 
34.5 

45.0 
44.4 
43.8 
43.2 
42.6 
42.0 
41.4 

52.5 

51.8 
51.1 
50.4 
49.7 
49.0 
48.3    . 

60.0 
59.2 
58.4 
57.6 
56.8 
56.0 
55.2 

67.5 
66.6 
65.7 
64.8 
63.9 
63.0 
62.1 

156 


LOGAK1THMS   OF   NUMBERS. 


No.  630  L.  799.1 


lNo.674L.829, 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

"630 

799341 

9409 

9478 

9547 

9616 

9685 

9754 

9823 

9892 

9961 

1 

800029 

0098 

0167 

0236 

0305 

0373 

0442 

0511 

0580 

0648 

2 

0717 

0786 

0854 

0923 

0992 

1061 

1129 

1198 

1266 

1335 

3 

1404 

1472 

1541 

1609 

1678 

1747 

1815 

1884 

1952 

2021 

4 

2089 

2158 

2226 

2295 

2363 

2432 

2500 

2568 

2637 

2705 

5 

2774 

2842 

2910 

2979 

3047 

3116 

3184 

3252 

3321 

3389 

6 

3457 

3525 

3594 

3662 

3730 

3798 

3867 

3935 

4003 

4071 

7 

4139 

4208 

4276 

4344 

4412 

4480 

4548 

4616 

4685 

4753 

8 

4821 

4889 

4957 

5025 

5093 

5161 

5229 

5297 

5365 

5433 

68 

9 

5501 

5569 

5637 

5705 

5773 

5841 

5908 

5976 

6044 

6112 

640 

806180 

6248 

6316 

6384 

6451 

6519 

6587 

6655 

6723 

6790 

1 

6858 

6926 

6994 

7061 

7129 

7197 

7264 

7332 

7400 

7467 

2 

7535 

7603 

7670 

7738 

7806 

7873 

7941 

8008 

8076 

8143 

3 

8211 

8279 

8346 

8414 

8481 

8549 

8616 

8684 

8751 

8818 

4 

8886 

8953 

9021 

9088 

9156 

9223 

9290 

9358 

9425 

9492 

5 

9560 

9627 

9694 

9762 

9829 

9896 

9964 

0031 

0098 

0165 

6 

810233 

0300 

0367 

0434 

0501 

0569 

0636 

0703 

0770 

0837 

7 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

6: 

8 

1575 

1642 

•1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

9 

2245 

2312 

2379 

2445 

2512 

2579 

2646 

2713 

2780 

2847 

650 

2913 

2980 

3047 

3114 

3181 

3247 

3314 

3381 

3448 

3514 

1 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

2 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

3 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

5378 

5445 

5511 

4 

5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

5 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

6 

6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

7 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

8 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

66 

9 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

660 

9544 

9610 

9676 

9741 

9807 

9873 

9939 

0004 

0070 

0136 

1 

820201 

0267 

0333 

0399 

0464 

0530 

0595 

0661 

0727 

0792 

2 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

3 

1514 

-1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

4 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

2626 

2691 

2756 

5 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

6 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3930 

3996 

4061 

7 

4126 

4191 

4256 

4321 

4386 

4451 

4516 

4581 

4646 

4711 

8 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

5361 

65 

9 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

670 

6075 

6140 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

1 

6723 

6787 

6852 

6917 

6981 

7046 

7111 

7175 

7240 

7305 

2 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

7951 

3 

8015 

8080 

8144 

8209 

8273 

8338 

8402 

8467 

8531 

8595 

4 

8660 

8724 

8789 

8853 

8918 

8982'  9046 

9111 

9175 

9239 

PROPORTIONAL  PARTS. 


Diff. 

1 

2 

3 

4 

5 

G 

7 

8 

9 

68 
67 
66 
65 
64 

6.8 
6.7 
6.6 
6.5 
6.4 

13.6 
13.4 
13.2 
13.0 
12.8 

20.4 
20.1 
19.8 
19.5 
19.2 

27.2 
26.8 
26.4 
26.0 
25.6 

34.0 
33.5 
33.0 
32.5 
32.0 

40.8 
40.2 
39.6 
39.0 

38.4 

47.6 
46.9 
46.2 
45.5 
44.8 

54.4 
53.6 
52.8 
52.0 
51.2 

61.2 
60.3 
59.4 
58.5 
57.6 

LOGARITHMS   Ofr  NtJMSERS. 


15? 


No.  675  L.  829.] 


[No.  719  L.  857. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

~67T 

829304 
9947 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9818 

9882 

6 

001  1 

0075 

0139 

0204 

0268 

0332 

0396 

0460 

"TTT 

7 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

3 

1230 

1294 

.  1358 

1422 

1486 

1550 

1614 

1678 

1742 

1806 

64 

9 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

680 

2509 

2573 

2637 

2700 

2764 

2828 

2892 

2956 

3020 

3083 

1 

3147 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

2 

3784 

3848 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

3 

4421 

4484 

4548 

4611 

4675 

.4739 

4802 

4866 

4929 

4993 

4 

5056 

5120 

5183 

5247 

5310 

5373 

5437 

5500 

5564 

5627 

5 

5691 

5754 

5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

6 

6324 

6387 

6451 

6514 

6577 

6641 

6704 

6767 

6830 

6894 

7 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

8 

7588 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

63 

9 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

690 

8849 

8912 

8975 

9038 

9101 

9164 

9227 

9289 

9352 

9415 

1 

94/8 

9541 

9604 

9667 

9729 

9792 

9855 

9918 

9981 

0043 

2 

840106 

0169 

0232 

0294 

0357 

0420 

0482 

0545 

0608 

0671 

3 

0733 

0796 

0859 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

4 

1359 

1422 

1485 

1547 

1610 

1672 

1735 

1797 

1860 

1922 

5 

1985 

2047 

2110 

2172 

2235 

2297 

2360 

2422 

2484 

2547 

6 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

3170 

7 

3233 

3295 

3357 

3420 

3482 

3544 

3606 

3669 

3731 

3793 

8 

3855 

3918 

3980 

4042 

4104 

4166 

4229 

4291 

4353 

4415 

9 

4477 

4539 

4601 

4664 

4726 

4788 

4850 

4912 

4974 

5036 

700 

5098 

5160 

5222 

5284 

5346 

5408 

5470 

5532 

5594 

5656 

62 

1 

5718 

5780 

5842 

5904 

5966 

6028 

6090 

6151 

6213 

6275 

2 

6337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

3 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

4 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

8128 

5 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

6 

8805 

8366 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

7 

9419 

9431 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

8 

850033 

0095 

0156 

0217 

0279 

0340 

0401 

0462 

0524 

0585 

9 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

710 

1258 

1320 

1381 

1442 

1503 

1564 

1625 

1686 

1747 

1809 

1 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

61 

2 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

3 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3516 

3577 

3637 

4 

3698 

3759 

3820 

3881 

3941 

4002 

4063 

4124 

4185 

4245 

5 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

6 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

7 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

8 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

9 

6729   6789J  6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

PROPORTIONAL  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

65 
64 
63 
62 
61 
60 

6.5 
6.4 
6.3 
6.2 
6.1 
6.0  1 

13.0 
12.8 
12.6 
12.4 
12.2 
12.0 

19.5 
19.2 
18.9 
18.6 
18.3 
18.0 

26.0 
25.6 
25.2 
24.8 
24.4 
24.0 

32.5 
32.0 
31.5 
31.0 
30.5 
30.0 

39.0 
38:4 
37.8 
37.2 
36.6 
36.0 

45.5 

44.8 
44.1 
43.4 
42.7 
42.0 

52.0 
51.2 
50.4 
49.6 
48.8 
48.0 

58.5 
57.6 
56.7 
55.8 
54.9 
54.0 

158 


LOGARITHMS   OF   NUMBERS. 


Ho.  720  L.  857.] 


(No.  764  L. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

720 

857332 

7393 

7453 

7513 

7574 

7634 

7694 

7755 

7815 

7875 

7935 

7995 

8056 

8116 

S176 

8236 

8297 

8357 

8417 

8477 

2 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

3 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

60 

4 

9739 

9799 

9859 

9918 

9978 

0038 

0098 

0158 

021fi 

fl?7fl 

5 

860338 

0398 

0458 

0518 

0578 

0637 

0697 

0757 

0817 

U^/O 

0877 

6 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

7 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

9 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3144 

3204 

3263 

730 

3323 

3382 

3442 

3501 

3561 

3620 

3680 

3739 

3799 

3858 

1 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

2 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

3 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

4 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

5 

6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

6 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

59 

7 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

8 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

9 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

9173 

740 

9232 

9290 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

9760 

1 

9818 

9877 

9935 

9994 

0053 

01  1  1 

0170 

0228 

0287 

0345 

2 

870404 

0462 

0521 

0579 

0638 

0696 

0755 

0813 

0872 

0930 

3 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

4 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2098 

5 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

6 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

3146 

3204 

3262 

7 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

8 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

58 

9 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

5003 

750 

5061 

5119 

5177 

5235 

5293 

5351 

5409 

5466 

5524 

5582 

1 

5640 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

2 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

3 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

- 

4 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

5 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

6 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

7 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

9669 

9726 

9784 

9841 

9898 

9956 

0013 

0070 

0127 

0185 

9 

880242 

0299 

0356 

0413 

0471 

0528 

0585 

0642 

0699 

0756 

760 

0814 

0871 

0928 

0985 

1042 

1099 

1156 

1213 

1271 

1328 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

2 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

57 

3 

2525 

2581 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

4 

3093 

3150 

3207 

3264 

3321 

3377 

3434 

3491 

3548 

3605 

PROPORTIONAL,  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

59 
58 
57 
56 

5.9 
5.8 
5.7 
5.6 

11.8 
11.6 
11.4 
11.2 

17.7 
17.4 
17.1 

16.8 

*23.6 
23.2 
22.8 
22.4 

29.5 
29.0 
28.5 
28.0 

35.4 
34.8 
34.2 
33.6 

41.3 
40.6 
39.9 
39.2 

47.2 
46.4 
45.6 
44.8 

53.1 
52.2 
51.3 
50.4 

LOGARITHMS   OF   NUMBERS. 


159 


No.  765  L.  883.] 


[No.  809  L.  908. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

765 

883661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

6 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4625 

4682 

4739 

7 

4795 

4852 

4909 

4965 

5022 

5078 

5135 

5192 

5248 

5305 

8 

5361 

5418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

9 

5926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

770 

6491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

1 

7054 

7111 

7167 

7223 

7280 

7336 

7392 

7449 

7505 

7561 

2 

7617 

7674 

7730 

7786 

7842 

-7898 

7955 

8011 

8067 

8123 

3 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8685 

A 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

9302 
9862 

9358 
9918 

9414 
9974 

9470 

9526 

9582 

9638 

9694 

9750 

9806 

56 

0030 

0086 

0141 

0197 

0253 

0309 

0365 

7 

890421 

0477 

0533 

0589 

0645 

0700 

0756 

0812 

0868 

0924 

8 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

9 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

780 

2095 

2150 

2206 

2262 

2317 

2373 

2429 

2484 

2540 

2595 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

2 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

3 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

4 

4316 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

5 

4870 

4925 

4980 

5036 

5091 

5146 

5201 

5257 

5312 

5367 

6 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

7 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

8 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

9 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

55 

790 

7627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

1 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

2 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

3 

9273 

9328 

9383 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

4 

9821 

9875 

9930 

9985 

0039 

0094 

0149 

0203 

0258 

0312 

5 

900367 

0422 

0476 

0531 

0586 

0640 

0695 

0749 

0804 

0859 

6 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

7 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

8 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

9 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

800 

3090 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

3524 

3578 

1 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

2 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

3 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

5202 

4 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

54 

5 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

6 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

7 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

8 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

9 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

PROPORTIONAL  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

57 
56 
55 
54 

5.7 
5.6 
5.5 
5.4 

11.4 
11.2 
11.0 
10.8 

17.1 
16.8 
16.5 
16.2 

22.8 
22.4 
22.0 
21.6 

28.5 
28.0 
27.5 
27.0 

34.2 
33.6 
33.0 
32.4 

39.9 
39.2 
38.5 
37.8 

45.6 
44.8 
44.0 
43.2 

51.3 
50.4 
49.5 
48.6 

LOGARITHMS    OF   NUMBERS. 


No.  810  L.  908.) 


[No.  854  L.  93* 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

810 

908485 

8539 

8592 

8646 

8699 

8753 

8807 

8860 

8914 

8967 

9021 

9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449 

9503 

2 

9556 

9610 

9663 

9716 

9770 

9823 

9877 

9930 

9984 

0037 

3 

910091 

0144 

0197 

0251 

0304 

0358 

0411 

0464 

0518 

0571 

4 

0624 

0678 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

5 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

6 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2116 

2169 

7 

2222 

2275 

2328 

2381 

2435 

2488 

2541 

2594 

2647 

2700 

8 

2753 

2806 

2859 

2913 

2966 

3019 

3072 

3125 

3178 

3231 

9 

3284 

3337 

3390 

3443 

3496 

3549 

3602 

3655 

3708 

3761 

53 

820 

3814 

3867 

3920 

3973 

4026 

4079 

4132 

4184 

4237 

4290 

1 

4343 

4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

2 

4872 

4925 

4977 

5030 

5083 

5136 

5189 

5241 

5294 

5347 

3 

5400 

5453 

5505 

5558 

5611 

5664 

5716 

5769 

5822 

5875 

4 

5927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

5 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

6 

6980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

7 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

8 

8030 

8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

9 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

830 

9078 

9130 

9183 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

1 

9601 

9653 

9706 

9758 

9810 

9862 

9914 

9Q67 

0019 

0071 

2 

920123 

0176 

0228 

0280 

0332 

0384 

0436 

0489 

0541 

0593 

3 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

5*- 

4 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

5 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

6 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

7 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

8 

3244 

3296 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

9 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

840 

4279 

4331 

4383 

4434 

4486 

4538 

4589 

4641 

4693 

4744 

1 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5261 

2 

5312 

5364 

5415 

5467 

5518 

5570 

5621 

5673 

5725 

5776 

3 

5828 

5879 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

4 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

5 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

6 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

7 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

8 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

9 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

850 

9419 

9470 

9521 

9572 

9623 

9674 

9725 

9776 

9827 

9879 

1 

9930 

9981 

51 

0032 

0083 

0134 

0185 

0236 

0287 

0338 

0389 

2 

930440 

0491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

3 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

4 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

PROPORTIONAL  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

» 

9 

53 
52 
51 
50 

5.3 
5.2 
5.1 
5.0 

10.6 
.10.4 
10.2 
10.0 

15.9 
15.6 
15.3 
15.0 

21.2 
20.8 
20.4 
20.0 

26.5 
26.0 
25.5 
25.0 

31.8 
31.2 
30.6 
30.0 

37.1 
36.4 
35.7 
35.0 

42.4 
41.6 
40.8 
40.0 

47.7 
46.8 
45.9 
45.0 

LOGARITHMS    OF   NUMBERS. 


161 


No.  855  L.  931. J 


[No.  899  L.  954, 


N. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

Diff. 

855 

931966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

6 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

7 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

8 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

9 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

860 

4498 

4549 

4599 

4650 

4700 

4751 

4801 

4852 

4902 

4953 

1 

5003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

2 

5507 

5558 

5608 

5658 

5709 

5759 

5809 

5860 

5910 

5960 

3 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

4 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

5 

7016 

7066 

7116 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

6 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

7 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

50 

6 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

9 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

870 

9519 

9569 

9619 

9669 

9719 

9769 

9819 

9869 

9918 

9968 

1 

940018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

0417 

0467 

2 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

3 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

4 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1958 

3 

2003 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

6 

2504 

2554 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

2950 

7 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

8 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

9 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

880 

4483 

4532 

4581 

4631 

4680 

4729 

4779 

4828 

4877 

4927 

1 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

2 

5469 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

5862 

5912 

3 

5961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

4 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

5 

6943 

6992 

7041 

7090 

7139 

7189 

7238 

7287 

7336 

7385 

6 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

49 

7 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

8 

8413 

8462 

8511 

8560 

8608 

8657 

8706 

8755 

8804 

8853 

9 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

690 

9390 

9439 

9488 

9536 

9585 

9634 

9683 

9731 

9780 

9829 

1 

9878 

9926 

9975 

0024 

0073 

0121 

0170 

0219 

0267 

0316 

2 

950365 

0414 

0462 

0511 

0560 

0608 

0657 

0706 

0754 

0303 

3 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

4 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

5 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

6 

2308 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

2696 

2744 

7 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

8 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

9 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

PROPORTIONAL  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

45.9 
45.0 
44.1 
43.2 

51 
50 
49 
48 

5.1 
5.0 
4.9 
48 

10.2 
10.0 
9.8 
9.6 

15.3 
15.0 
14.7 
14.4 

20.4 
20.0 
19.6 
19.2 

25.5 
25.0 
24.5 
24.0 

30.6 
30.0 
29.4 
28.8 

35.7 
35.0 
34.3 
33.6 

40.8 
40.0 
39.2 
38.4 

162 


LOGARITHMS    OF    NUMBERS. 


Ho.  900  L.  954.J 


[No.  944  L.  975. 


N. 

1   0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

900 

954243 

4291 

4339 

4387 

4435 

4484 

4532 

4580 

4628 

4677 

. 

1 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

5110 

5158 

2 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

3 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

4 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6553 

6601 

5 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48 

6 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7559 

7 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

8 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

9 

8564 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

910 

9041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

9471 

I 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

2 

9995 

0042 

0090 

Atao 

ni  oe 

n?^ 

A7Af 

rjaoo 

n^7A 

3 

960471 

0518 

0566 

V  1  JO 

0613 

U  1  O  J 

0661 

\jLjj 

0709 

UZOU 

0756 

\)jZ,c 

0804 

U.3/O 

0851 

0423 
0899 

4 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

5 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

6 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

7 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

8 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

9 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

1 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

2 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

5108 

5155 

3 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5625 

4 

5672 

5719 

5766 

5813 

5860 

5907 

5954 

6001 

6048 

6095 

47 

5 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

6 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

7 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

8 

7548 

7595 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

7969 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

930 

8483 

8530 

8576 

8623 

8670 

8716 

8763 

8810 

8856 

8903 

1 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

2 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

3 

9882 

9928 

9975 

0021 

0068 

01  14 

0161 

0207 

0254 

0300 

4 

970347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

5 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

6 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

7 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2064 

2110 

2157 

8 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

9 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

940 

3128 

3174 

3220 

3266 

3313 

3359 

3405 

3451 

3497 

3543 

1 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

2 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

3 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

4 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

46 

PROPORTIONAL  PARTS. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

47 
46 

4.7 
4.6 

9.4 
9.2 

14.1 
13.8 

18.8 
18.4 

23.5 
23.0 

28.2 
27.6 

32.9 
32.2 

37.6 
36.8 

42.3 
41.4 

LOGARITHMS    OF   NUMBERS. 


163 


No.  945  L.  975.J 


(No.  989  L.  995. 


N. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

Diff. 

945 

975432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

6 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

7 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

8 

6808 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

9 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

950 

7724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

1 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

2 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

3 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

4 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

5 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

6 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

7 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

8 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

9 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

960 

2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

1 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

2 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

3 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

4 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

5 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

6 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

7 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

5741 

5786 

5830 

8 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

9 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

6772 

6817 

6861 

6906 

6951 

6996 

7040 

7085 

7130 

7175 

1 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

2 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

3 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

5 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

7 
7 

9395 

9939 

9983 

0028 

0072 

01  17 

0161 

0206 

025C 

O7O4 

8 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

U^Vn 

0738 

9 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

1 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

2 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

3 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

4 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

5 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

6 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

7 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

44 

8 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

9 

5196 

5240 

5284 

5328 

5372 

5416 

5460 

5504 

5547 

5591 

PROPORTIONAL  PARTS. 


Diff. 

46 
45 
44 
43 

1 

3 

3 

4 

5 

6 

7 

8 

9 

4.6 
4.5 
4.4 
4.3 

9.2 
9.0 
8.8 
8.6 

13.8 
13.5 
13.2 
12.9 

18.4 
18.0 
17.6 
17.2 

23.0 
22.5 
22.0 
21.5 

27.6 
27.0 
26.4 
25.8 

32.2 
31.5 
30.8 
30.1 

36.8 
36.0 
35.2 
34.4 

41.4 
40.5 
39.6 
38.7 

164 


HYPERBOLIC   LOGARITHMS. 


No. 990  L. 995 J 


[No.999L.999. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

990 

995635 

5679 

5723 

5767 

5811 

5854 

5898 

5942 

5986 

6030 

1 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

44 

2 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

3 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

4 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

3 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

6 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

7 

8695 

8739 

8782 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

8 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9522 

9 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

43 

HYPERBOLIC  LOGARITHMS. 


No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

1.01 

.0099 

1.45 

.3716 

.89 

.6366 

2.33 

.8458 

2.77 

.0188 

1.02 

.0198 

1.46 

.3784 

.90 

.6419 

2.34 

.8502 

2.78 

.0225 

1.03 

.0296 

1.47 

.3853 

.91 

.6471 

2.35 

.8544 

2.79 

.0260 

1.04 

.0392 

1.48 

.3920 

.92 

.6523 

2.36 

.8587 

2.80 

.0296 

1.05 

.0488 

1.49 

.3988 

.93 

.6575 

2.37 

.8629 

2.81 

.0332 

1.06 

.0583 

1.50 

.4055 

.94 

.6627 

2.38 

.8671 

2.82 

.0367 

1.07 

.0677 

1.51 

.4121 

.95 

.6678 

2.39 

.8713 

2.83 

.0403 

1.08 

.0770 

1.52 

.4187 

.96 

.6729 

2.40 

.8755 

2.84 

.0438 

1.09 

.0862 

1.53 

.4253 

.97 

.6780 

2.41 

.8796 

2.85 

.0473 

1.10 

.0953 

1.54 

.4318 

.98 

.6831 

2.42 

.8838 

2.86 

.0508 

1.11 

.1044 

1.55 

.4383 

1.99 

.6881 

2.43 

.8879 

2.87 

.0543 

1.12 

.1133 

1.56 

.4447 

2.00 

.6931 

2.44 

.8920 

2.88 

.0578 

1.13 

.1222 

1.57 

.4511 

2.01 

.6981 

2.45 

.8961 

2.89 

.0613 

1.14 

.1310 

1.58 

.4574 

2.02 

.7031 

2.46 

.9002 

2.90 

.0647 

1.15 

.1398 

1.59 

.4637 

2.03 

.7080 

2.47 

.9042 

2.91 

.0682 

1.16 

.1484 

1.60 

.4700 

2.04 

.7129 

2.48 

.9083 

2.92 

.0716 

1.17 

.1570 

1.61 

.4762 

2.05 

.7178 

2.49 

.9123 

2.93 

.0750 

1.18 

.1655 

1.62 

.4824 

2.06 

.7227 

2.50 

.9163 

2.94 

.0784 

1.19 

.1740 

1.63 

.4886 

2.07 

.7275 

2.51 

.9203 

2.95 

.0818 

1.20 

.1823 

1.64 

.4947 

2.08 

.7324 

2.52 

.9243 

2.96 

.0852 

1.21 

.1906 

1.65 

.5008 

2.09 

.7372 

2.53 

.9282 

2.97 

.0886 

1.22 

.1988 

J.66 

.5068 

2.10 

.7419 

2.54 

.9322 

2.98 

.0919 

1.23 

.2070 

1.67 

.5128 

2.11 

.7467 

2.55 

.9361 

2.99 

.0953 

1.24 

.2151 

1.68 

.5188 

2.12 

.7514 

2.56 

.9400 

3.00 

.0986 

1.25 

.2231 

1.69 

.5247 

2.13 

.7561 

2.57 

.9439 

3.01 

.1019 

1.26 

.2311 

1.70 

.5306 

2.14 

.7608 

2.58 

.9478 

3.02 

.1056 

1.27 

.2390 

1.71 

.5365 

2.15 

.7655 

2.59 

.9517 

3.03 

.1081 

1.28 

.2469 

1.72 

.5423 

2.16 

.7701 

2.60 

.9555 

3.04 

.1113 

1.29 

.2546 

1.73 

.5481 

2.17 

.7747 

2.61 

.9594 

3.05 

.1154 

1.30 

.2624 

1.74 

.5539 

2.18 

.7793 

2.62 

.9632 

3.06 

.1187 

1.31 

.2700 

1.75 

.5596 

2.19 

.7839 

2.63 

.9670 

3.07 

.1219 

1.32 

.2776 

1.76 

.5653 

2.20 

.7885 

2.64 

.9708 

3.08 

.1246 

1.33 

.2852 

1.77 

.5710 

2.21 

.7930 

2.65 

.9746 

3.09 

.1284 

1.34 

.2927 

1.78 

.5766 

2.22 

.7975 

2.66 

.9783 

3.10 

.1312 

1.35 

.3001 

1.79 

.5822 

2.23 

.8020 

2.67 

.9821 

3.11 

.1349 

1.36 

.3075 

1.80 

.5878 

2.24 

.8065 

2.68 

.9858 

3.12 

.1378 

1.37 

.3148 

1.81 

.5933 

2.25 

.8109 

2.69 

.9895 

3.13 

.1410 

1.38 

.3221 

1.82 

.5988 

2.26 

.8154 

2.70 

.9933 

3.14 

.1442 

1.39 

.3293 

1.83 

.6043 

2.27 

.8198 

2.71 

.9969 

3.15 

.1474 

1.40 

.3365 

1.84 

.6098 

2.28 

.8242 

2.72 

1  .0006 

3.16 

.1506 

1.41 

.3436 

1.85 

.6152 

2.29 

.8286 

2.73 

1  .0043 

3.17 

.1537 

1.42 

.3507 

1.86 

.6206 

2.30 

.8329 

2.74 

1  .0080 

3.18 

.1569 

1.43 

.3577 

1.87 

.6259 

2.31 

.8372 

2.75 

1.0116 

3.19 

.1600 

1.44 

.3646 

1.88 

.6313 

2.32 

.8416 

2.76 

1.0152 

3.20 

.1632 

HYPERBOLIC  LOGARITHMS. 


165 


No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

3.21 

.1663 

3.87 

.3533 

4.53 

.5107 

5.19 

.6467 

5.85 

.7664 

3.22 

.1694 

3.88 

.3558 

4.54 

.5129 

5.20 

.6487 

5.86 

.7681 

3.23 

.1725 

3.89 

.3584 

4.55 

.5151 

5.21 

.6506 

5.87 

.7699 

3.24 

.1756 

3.90 

.3610 

4.56 

.5173 

5.22 

.6525 

5.88 

.7716 

3.25 

.1787 

3.91 

.3635 

4.57 

.5195 

5.23 

.6544 

5.89 

.7733 

3.26 

.1817 

3.92 

.3661 

4.58 

.5217 

5.24 

.6563 

5.90 

.7750 

3.27 

.1848 

3.93 

.3686 

4.59 

.5239 

5.25 

.6582 

5.91 

.7766 

3.28 

.1878 

3.94 

.3712 

4.60 

.5261 

5.26 

.6601 

5.92 

.7783 

3.29 

.1909 

3.95 

.3737 

4.61 

.5282 

5.27 

.6620 

5.93 

.7800 

3.30 

.1939 

3.96 

.3762 

4.62 

.5304 

5.28 

.6639 

5.94 

.7817 

3.31 

.1969 

3.97 

.3788 

4.63 

.5326 

5.29 

.6658 

5.95 

.7834 

3.32 

.1999 

3.98 

.3813 

4.64 

.5347 

5.30 

.6677 

.  5.96 

.7851 

3.33 

.2030 

3.99 

.3838 

4.65 

.5369 

5.31 

.6696 

5.97 

.7867 

3.34 

.2060 

4.00 

.3863 

4.66 

.5390 

5.32 

.6715 

5.98 

.7884 

3.35 

.2090 

4.01 

.3888 

4.67 

.5412 

5.33 

.6734 

5.  99' 

.7901 

3.36 

.2119 

4.02 

.3913 

4.68 

.5433 

5.34 

.6752 

6.00 

.7918 

3.37 

.2149 

4.03 

.3938 

4.69 

.5454 

5.35 

.6771 

6.01 

.7934 

3.38 

.2179 

4.04 

.3962 

4.70 

.5476 

5.36 

.6790 

6.02 

.7951 

3.39 

.2208 

4.05 

.3987 

4.71 

.5497 

5.37 

.6808 

6.03 

.7967 

3.40 

.2238 

4.06 

.4012 

4.72 

.5518 

5.38 

.6827 

6.04 

.7984 

3.41 

.2267 

4.07 

.4036 

4.73 

.5539 

5.39 

.6845 

6.05 

.8001 

3.42 

.2296 

4.08 

.4061 

4.74 

.5.560 

5.40 

.6864 

606 

.8017 

3.43 

.2326 

4.09 

.4085 

4.75 

.5581 

5.41 

.6882 

6.07 

.8034 

3.44 

.2355 

4.10 

.4110 

4.76 

.5602 

5.42 

.6901 

6.08 

.8050 

3.45 

.2384 

4.11 

.4134 

4.77 

.5623 

5.43 

.6919 

6.09 

.8066 

3.46 

.2413 

4.12 

.4159 

4.78 

.5644 

5.44 

.6938 

6.10 

.8083 

3.47 

.2442 

4.13 

.4183 

4.79 

.5665 

5.45 

.6956 

6.11 

.8099 

3.48 

.2470 

4.14 

.4207 

4.80 

.5686 

5.46 

.6974 

6.12 

.8116 

3.49 

.2499 

4.15 

.4231 

4.81 

.5707 

5.47 

.6993 

6.13 

.8132 

3.50 

.2528 

4.16 

.4255 

4.82 

.5728 

5.48 

.7011 

6.14 

.8148 

3.51 

.2556 

4.17 

.4279 

4.83 

.5748 

5.49 

.7029 

6.15 

.8165 

3.52 

.2585 

4.18 

.4303 

4.84 

.5769 

5.50 

.7047 

6.16 

.8181 

3.53 

.2613 

4.19 

.4327 

4.85 

.5790 

5.51 

.7066 

6.17 

.8197 

3.54 

.2641 

4.20 

.4351 

4.86 

.5810 

5.52 

.7084 

6.18 

.8213 

3.55 

.2669 

4.21 

.4375 

4.87 

.5831 

5.53 

.7102 

6.19 

.8229 

3.56 

.2698 

4.22 

.4398 

4.88 

5851 

5.54 

.7120 

6.20 

.8245 

3.57 

.2726 

4.23 

.4422 

4.89 

.5872 

5.55 

.7138 

6.21 

.8262 

3.58 

.2754 

4.24 

4446 

4.90 

.5892 

5.56 

.7156 

6.22 

.8278 

3.59 

.2782 

4.25 

.4469 

4.91 

.5913 

5.57 

.7174 

6.23 

.8294 

3.60 

.2809 

4.26 

.4493 

4.92 

.5933 

5.58 

.7192 

6.24 

.8310 

3.61 

.2837 

4.27 

.4516 

4.93 

.5953 

5.59 

.7210 

6.25 

.8326 

3.62 

.2865 

4.28 

.4540 

4.94 

.5974 

5.60 

.7228 

6.26 

.8342 

3.63 

.2892 

4.29 

.4563 

4.95 

.5994 

5.61 

.7246 

6.27 

.8358 

3.64 

.2920 

4.30 

.4586 

4.96 

.6014 

5.62 

.7263 

6.28 

.8374 

365 

.2947 

4.31 

.4609 

4.97 

.6034 

5.63 

.7281 

6.29 

.8390 

3.66 

.2975 

4.32 

.4633 

4.98 

.6054 

5.64 

.7299 

6.30 

.8405 

3.67 

.3002 

4.33 

.4656 

4.99 

.6074 

5.65 

.7317 

6.31 

.8421 

3.68 

.3029 

4.34 

.4679 

5.00 

.6094 

5.66 

.7334 

6.32 

.8437 

3.69 

.3056 

4.35 

.4702 

5.01 

.6114 

5.67 

.7352 

6.33 

.8453 

3.70 

.3083 

4.36 

.4725 

5.02 

.6134 

5.68 

.7370 

6.34 

8469 

3.71 

.3110 

4.37 

.4748 

5.03 

.6154 

5.69 

.7387 

6.35 

.8485 

3.72 

.3137 

4.38 

.4770 

5.04 

.6174 

5.70 

.7405 

6.36 

.8500 

3.73 

.3164 

4.39 

.4793 

5.05 

.6194 

5.71 

.7422 

6.37 

.8516 

3.74 

.3191 

4.40 

.4816 

5.06 

.6214 

5.72 

.7440 

6.38 

.8532 

3.75 

.3218 

4.41 

.4839 

5.07 

.6233 

5.73 

.7457 

6.39 

.8547 

3.76 

.3244 

4.42 

.4861 

5.08 

.6253 

5.74 

.7475 

6.40 

.8563 

3.77 

.3271 

4.43 

.4884 

5.09 

.6273 

5.75 

.7492 

6.41 

.8579 

3.78 

.3297 

4.44 

.4907 

5.10 

.6292 

5.76 

.7509 

6.42 

.8594 

3.79 

.3324 

4.45 

.4929 

5.11 

.6312 

5.77 

.7527 

6.43 

.8610 

3.80 

.3350 

4.46 

.4951 

5.12 

.6332 

5.78 

.7544 

6.44 

.8625 

3.81 

.3376 

4.47 

.4974 

5.13 

.6351 

5.79 

.7561 

6.45 

.8641 

3.82 

.3403 

4.48 

.4996 

5.14 

.6371 

5.80 

.7579 

6.46 

.8656 

3.83 

.3429 

4.49 

.5019 

5.15 

.6390 

5.81 

.7596 

6.47 

.8672 

3.84 

.3455 

4.50 

.5041 

5.16 

.6409 

5.82 

.7613 

6.48 

8687 

3.85 

.3481 

4.51 

.5063 

5.17 

.6429 

5.83 

1  .7630 

6.49 

.8703 

3.86 

.3507 

4.52 

.5085 

5.18 

.6448 

5.84 

1.7647 

6.50 

.8718 

106 


HYPERBOLIC  LOGARITHMS. 


No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

No 

Log. 

6.51 

1.8733 

7.15 

.9671 

7.79 

2.0528 

8.66 

2.1587 

9.94 

2.2966 

6.52 

1.8749 

7.16 

.9685 

7.80 

2.0541 

8.68 

2.1610 

9.96 

2.2986 

6.53 

1  .8764 

7.17 

.9699 

7.81 

2.0554 

8.70 

2.1633 

9.98 

2.3006 

6.54 

1.8779 

7.18 

.9713 

7.82 

2.0567 

8.72 

2  1656 

10.00 

2.3026 

6.55 

1.8795 

7.19 

.9727 

7.83 

2.0580 

8.74 

2.1679 

10.25 

2.3279 

6.56 

1.8810 

7.20 

.9741 

7.84 

2.0592 

8.76 

2.1702 

10.50 

2.3513 

6.57 

1.8825 

7.21 

.9754 

7.85 

2.0605 

8.78 

2.1725 

10.75 

2.3749 

6.58 

1  .8840 

7.22 

.9769 

7.86 

2.0618 

8.80 

2.1748 

11.00 

2.3979 

6.59 

1.8856 

7.23 

.9782 

7.87 

2.0631 

8.82 

2.1770 

11.25 

2.4201 

6.60 

1.8871 

7.24 

.9796 

7.88 

2.0643 

8.84 

2.1793 

11.50 

2.4430 

6.61 

1  .8886 

7.25 

.9810 

7.89 

2.0656 

8.86 

2.1815 

11.75 

2.4636 

6.62 

1.8901 

7.26 

.9824 

7.90 

2.0669 

8.88 

2.1838 

12.00 

2.4849 

6.63 

1.8916 

7.27 

.9838 

7.91 

2.0681 

8.90 

2.1861 

12.25 

2.5052 

6.64 

1  .893  1 

7.28 

.9851 

7.92 

2.0694 

8.92 

2.1883 

12.50 

2.5262 

6.65 

1  .8946 

7.29 

.9865 

7.93 

2.0707 

8.94 

2.1905 

12.75 

2.5455 

6.66 

1  8961 

7.30 

.9879 

7.94 

2.0719 

8.96 

2.1928 

13.00 

2.5649 

6.67 

1.8976 

7.31 

.9892 

7.95 

2.0732 

8.98 

2.1950 

13.25 

2.5840 

6.68 

1.8991 

7.32 

.9906 

7.96 

2.0744 

9.00 

2.1972 

13.50 

2.6027 

6.69 

1  .9006 

7.33 

.9920 

7.97 

2.0757 

9.02 

2.1994 

13.75 

2.621  1 

6.70 

1.9021 

7.34 

.9933 

7.98 

2.0769 

9.04 

2.2017 

14.00 

2.6391 

6.71 

1  .9036 

7.35 

.9947 

7.99 

2.0782 

9.06 

2.2039 

14.25 

2.6567 

6.72 

1.9051 

7.36 

.9961 

800 

2.0794 

9.08 

2.2061 

14.50 

2.6740 

6.73 

1  .9066 

7.37 

.9974 

8.01 

2.0807 

9.10 

2.2083 

14.75 

2.6913 

6.74 

1.9081 

7.38 

.9988 

8.02 

2.0819 

9.12 

2.2105 

15.00 

2.7081 

6.75 

1  .9095 

7.39 

2.0001 

8.03 

2.0832 

9.14 

2.2127 

15.50 

2.7408 

6.76 

1.9110 

7.40 

2.0015 

8.04 

2.0844 

9.16 

2.2148 

16.00 

2.7726 

6.77 

1.9125 

7.41 

2.0028 

8.05 

2.0857 

9.18 

2.2170 

16.50 

2.8034 

6.78 

1.9140 

7.42 

2.0041 

8.06 

2.0869 

9.20 

2.2192 

17.00 

2.8332 

6.79 

1.9155 

7.43 

2.0055 

8.07 

2.0882 

9.22 

2.2214 

17.50 

2.8621 

6.80 

1.9169 

7.44 

2.0069 

8.08 

2.0894 

924 

2.2235 

18.00 

2.8904 

6.81 

1.9184 

7.45 

2.0082 

8.09 

2.0906 

9.26 

2.2257 

18.50 

2.9178 

6.82 

1.9199 

7.46 

2.0096 

8.10 

2.0919 

9.28 

2.2279 

19.00 

2.9444 

6.83 

1.9213 

7.47 

2.0108 

8.11 

2.0931 

9.30 

2.2300 

19.50 

2.9703 

6.84 

1  .9228 

7.48 

2.0122 

8.12 

2.0943 

9.32 

2.2322 

20.00 

2.9957 

6.85 

1  .9242 

7.49 

2.0136 

8.13 

2.0956 

9.34 

2.2343 

21 

3.0445 

6.86 

1.9257 

7.50 

2.0149 

8,14 

2.0968 

9.36 

2.2364 

22 

3.0910 

6.87 

1.9272 

7.51 

2.0162 

8.15 

2.0980 

9.38 

2.2386 

23 

3.1355 

6.88 

1  .9286 

7.52 

2.0176 

8.16 

2.0992 

9.40 

2.2407 

24 

3.1781 

6.89 

1.9301 

7.53 

2.0189 

8.17 

2.1005 

9:42 

2.2428 

25 

3.2189 

6.90 

1  .93  1  5 

7.54 

2.0202 

8.18 

2.1017 

9.44 

2.2450 

26 

3.2581 

N6.91 

1  .9330 

7.55 

2.0215 

8.19 

2.1029 

9.46 

2.2471 

27 

3.2958 

6.92 

1.9344 

7.56 

2.0229 

8.20 

2.1041 

9.48 

2.2492 

28 

3.3322 

6.93 

1.9359 

7.57 

2.0242 

8.22 

2.1066 

9.50 

2.2513 

29 

3.3673 

6.94 

1.9373 

7.58 

2.0255 

8.24 

2.1090 

9.52 

2.2534 

30 

3.4012 

6.95 

1.9387 

7.59 

2.0268 

8.26 

2.1114 

9.54 

2.2555 

31 

3.4340 

6.96 

1  .9402 

7.60 

2.0281 

8.28 

2.1138 

9.56 

2.2576 

32 

3.4657 

6.97 

1.9416 

7.61 

2.0295 

8.30 

2.1163 

9.58 

2.2597 

33 

3.4965 

6.98 

1  .9430 

7.62 

2.0308 

8.32 

2.1187 

9.60 

2.2618 

34 

3.5263 

6.99 

1  .9445 

7.63 

2.0321 

8.34 

2.1211 

9.62 

2.2638 

35 

3.5553 

7.00 

1  .9459 

7.64 

2.0334 

8.36 

2.1235 

9.64 

2.2659 

36 

3.5835 

7.01 

1  .9473 

7.65 

2.0347 

8.38 

2.1258 

9.66 

2.2680 

37 

3.6109 

7.02 

1  .9488 

7.66 

2.0360 

8.40 

2.1282 

9.68 

2.2701 

38 

3.6376 

7.03 

1  .9502 

7.67 

2.0373 

8.42 

2.1306 

9.70 

2.2721 

39 

3.6636 

7.04 

1.9516 

7.68 

2.0386 

8.44 

2.1330 

9.72 

2.2742 

40 

3.6889 

7.05 

1.9530 

7.69 

2.0399 

8.46 

2.1353 

9.74 

2.2762 

41 

3.71J6 

7.06 

1.9544 

7.70 

2.0412 

8.48 

2.1377 

9.76 

2.2783 

42 

3.7377 

7.07 

1.9559 

7.71 

2.0425 

8.50 

2.1401 

9.78 

2.2803 

43 

3.7612 

7.08 

1.9573 

7.72 

2.0438 

8.52 

2.1424 

9.80 

2.2824 

44 

3.7842 

7.09 

1.9587 

7.73 

2.0451 

8.54 

2.1448 

9.82 

2.2844 

45 

3.8067 

7.10 

1.9601 

7.74 

2.0464 

8.56 

2.1471 

9.84 

2.2865 

46 

3.8286 

7.11 

1.9615 

7.75 

2.0477 

8.58 

2.1494 

9.86 

2.2885 

47 

3.8501 

7.12 

1  .9629 

7.76 

2.0490 

8.60 

2.1518 

9.88 

2.2905 

48 

3.8712 

7.13 

1  .9643 

7.77 

2.0503 

8.62 

2.1541 

9.90 

2.2925 

49 

3.8918 

7.14 

1.9657 

7.78 

2.0516 

8.64 

2.1564 

9.92 

2.2946 

50 

3.9120 

LOGARITHMIC    TRIGONOMETRICAL   FUNCTIONS. 


167 


LOGARITHMIC   SINES,  ETC. 


1 

Sine. 

Cosec. 

Versin. 

Tangent 

Cotan. 

Covers. 

Secant. 

Cosine. 

bb 

0) 

Q 

o 

n.Neg. 

nfinite. 

n.Neg. 

In.Neg. 

Infinite. 

0.00000 

1  0.00000 

0.00000 

90 

1 

.24186 

1.75814 

6.18271 

8.24192 

11.75808 

9.99235 

10.00007 

9.99993 

89 

2 

.54282 

1.43718 

6.78474 

8.54308 

1  1  .45692 

9.98457 

10.00026 

9.99974 

88 

3 

.71880 

1.28120 

7.13687 

8.71940 

1  1  .28060 

9.97665 

10.00060 

9.99940 

87 

4 

.84358 

1.15642 

7.38667 

8.84464 

11.15536 

9.96860 

10.00106 

9.99894 

80 

5 

.94030 

1.05970 

7.58039 

8.94195 

1  1  .05805 

9.96040 

10.00166 

9.99834 

85 

6 

.01923 

0.98077 

7.73863 

9.02162 

10.97838 

9.95205 

10.00239 

999761 

84 

7 

9.08589 

0.91411 

7.87238 

9.08914 

10.91086 

9.94356 

10.00325 

9.99675 

83 

8 

.14356 

0.85644 

7.98820 

9.14780 

10.85220 

9.93492 

10.00425 

9.99575 

82 

9 

.19433 

0.80567 

8.09032 

9.19971 

10.80029 

9.92612 

0.00538 

9.99462 

81 

10 

9.23967 

0.76033 

8.18162 

9.24632 

10.75368 

9.91717 

10.00665 

9.99335 

80 

11 

9.28060 

0.71940 

8.26418 

9.28865 

10.71135 

9.90805 

10.00805 

9.99195 

79 

12 

9.31788 

0.68212 

8.33950 

9.32747 

10.67253 

9.89877 

10.00960 

9.99040 

78 

13 

9.35209 

0.64791 

8.40875 

9.36336 

0.63664 

9.88933 

10.01128 

9.98872 

77 

14 

9.38368 

0.61632 

8.47282 

9.39677 

0.60323 

9.87971 

10.01310 

9.98690 

76 

15 

9.41300 

0.58700 

8.53243 

9.42805 

10.57195 

9.86992 

10.01506 

9.98494 

75 

16 

9.44034 

0.55966 

8.58814 

9.45750 

10.54250 

9.85996 

10.01716 

9.98284 

74 

17 

9.46594 

0.53406 

8.64043 

9  48534 

10.51466 

9.84981 

10.01940 

9.98060 

73 

18 

9  48998 

0.51002 

8.68969 

9.51178 

10.48822 

9.83947 

10.02179 

9.97821 

72 

19 

9.51264 

10.48736 

8.73625 

9.53697 

10.46303 

9.82894 

10.02433 

9.97567 

71 

20 

9.53405 

10.46595 

8.78037 

9.56107 

10.43893 

9.81821 

10.02701 

9.97299 

70 

21 

9.55433 

10.44567 

8.82230 

9.58418 

10.41582 

9.80729 

10.02985 

9.97015 

69 

22 

9.57358 

10.42642 

8.86223 

9.60641 

10.39359 

9.79615 

10.03283 

9.96717 

68 

23 

9.59188 

10.40812 

8.90034 

9.62785 

10.37215 

9.78481 

10.03597 

9.96403 

67 

24 

9.6093  1 

10.39069 

8.93679 

9.64858 

10.35142 

9.77325 

10.03927 

9.96073 

66 

25 

9.62595 

10.37405 

8.97170 

9.66867 

10.33133 

9.76146 

10.04272 

9.95728 

65 

26 

9.64184 

10.35816 

9.00521 

9.68818 

10.31182 

9.74945 

10.04634 

9.95366 

64 

27 

9.65705 

10.34295 

9.03740 

9.70717 

10.29283 

9.73720 

10.05012 

9.94988 

63 

28 

9.67161 

10.32839 

9.06838 

9.72567 

10.27433 

9.72471 

10.05407 

9.94593 

62 

29 

9.68557 

10.31443 

9.09823 

9.74375 

10.25625 

9.71197 

10.05818 

9.94182 

61 

30 

9.69897 

10.30103 

9.12702 

9.76144 

10.23856 

9.69897 

10.06247 

9.93753 

60 

31 

9.71184 

10.28816 

9.15483 

9.77877 

10.22123 

9.68571 

10.06693 

9.93307 

59 

32 

9.72421 

10.27579 

9.18171 

9.79579 

10.20421 

9.67217 

10.07158 

9.92842 

58 

33 

9.73611 

10.26389 

9.20771 

9.81252 

10.18748 

9.65836 

10.07641 

9.92359 

57 

34 

9.74756 

10.25244 

9.23290 

9.82899 

10.17101 

9.64425 

10.08143 

9.91857 

56 

35 

9.75859 

10.24141 

9.25731 

9.84523 

10.15477 

9.62984 

10.08664 

9.91336 

55 

36 

9.76922 

10.23078 

9.28099 

9.86126 

10.13874 

9.61512 

10.09204 

9.90796 

54 

37 

9.77946 

10.22054 

9.30398 

9.87711 

10.12289 

9.60008 

10.09765 

9.90235 

53 

38 

9.78934 

10.21066 

9.32631 

9.89281 

10.10719 

9.58471 

10.10347 

9.89653 

52 

39 

9.79887 

10.20113 

9.34802 

9.90837 

10.09163 

9.56900 

10.10950 

9.89050 

51 

40 

9.80807 

10.19193 

9.36913 

9.92381 

10.07619 

9.55293 

10.11575 

9.88425 

50 

41 

9.81694 

10.18306 

9.38968 

9.93916 

10.06084 

9.53648 

10.12222 

9.87778 

49 

42 

9.82551 

10.17449 

9.40969 

9.95444 

10.04556 

9.51966 

10.12893 

9.87107 

48 

43 

9.83378 

10.16622 

9.42918 

9.96966 

10.03034 

9.50243 

10.13587 

9.86413 

47 

44 

9.84177 

10.15823 

9.44818 

9.98484 

10.01516 

9.48479 

10.14307 

9.85693 

46 

45 

9.84949 

10.15052 

9.4667 

10.00000 

10.00000 

9.46671 

10.15052 

9.84949 

45 

Cosine 

Secant. 

Covers 

Cotan. 

Tangent 

Versin. 

Cosec. 

Sine. 

From  45°  to  90°  read  from  bottom  of  table  upwards. 


168 


LOGARITHMS   OF  NUMBERS. 


Four-place  Logarithms  of  Numbers  to   1000. 

For  six-place  logarithms  of  numbers  to  10,000,  see  pp.  137  to  164. 


No. 

0 

1 

2 

3 

4 

5 

6 

1 

8 

9 

No. 

0 

0000 

3010 

4771 

6021 

6990 

7782 

8451 

9031 

9542 

0 

2 
3 

4 
5 
6 
7 
8 
9 

0000 
3010 
4771. 
6021 
6990 
7782 
8451 
9031 
9542 

0414 
3222 
4914 
6128 
7076 
7853 
8513 
9085 
9590 

0792 
3424 
5052 
6232 
7160 
7924 
8573 
9138 
9638 

1139 
3617 
5185 
6335 
7243 
7993 
8633 
9191 
9685 

1461 
3802 
5315 
6435 
7324 
8062 
8692 
9243 
9731 

1761 
3979 
5441 
6532 
7404 
8129 
8751 
9294 
9777 

2041 
4150 
5563 
6628 
7482 
8195 
8808 
9345 
9823 

2304 
4314 
5682 
6721 
7559 
8261 
8865 
9395 
9868 

2553 
4472 
5798 
6812 
7634 
8325 
8921 
9445 
9912 

2788 
4624 
5911 
6902 
7709 
8388 
8976 
9494 
9956 

1 
2 
3 
4 
5 
6 
7 
8 
9 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

10 

It 
12 
13 
14 
15 
16 
17 
18 
19 

0414 
0792 
1139 
1461 
1761 
2041 
2304 
2553 
2788 

0453 
0828 
1173 
1492 
1790 
2068 
2330 
2577 
2810 

0492 
0864 
1206 
1523 
1818 
2095 

2355 
2601 
2833 

0531 
0899 
1239 
1553 
1847 
2122 
2380 
2625 
2856 

0569 
0934 
1271 
1584 
1875 
2148 
2405 
2648 
2878 

0607 
0969 
1303 
1614 
1903 
2175 
2430 
2672 
2900 

0645 
1004 
1335 
1644 
1931 
2201 
2455 
2695 
2923 

0682 
1038 
1367 
1673 
1959 
2227 
2480 
2718 
2945 

0719 
1072 
1399 
1703 
1987 
2253 
2504 
2742 
2967 

0755 
1106 
1430 
1732 
2014 
2279 
2529 
2765 
2989 

11 

12 
13 
14 
15 
16 
17 
18 
19 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 

3222 

3424 
3617 
3802 
3979 
4150 
4314 
4472 
4624 

3243 
3444 
3636 
3820 
3997 
4166 
4330 
4487 
4639 

3263 
3464 
3655 
3838 
4014 
4183 
4346 
4502 
4654 

3284 
3483 
3674 
3856 
4031 
4200 
4362 
4518 
4669 

3304 
3502 
3692 
3874 
4048 
4216 
4378 
4533 
4683 

3324 
3522 
3711 
3892 
4065 
4232 
4393 
4548 
4698 

3345 
3541 
3729 
3909 
4082 
4249 
4409 
4564 
4713 

3365 
3560 
3747 
3927 
4099 
4265 
4425 
4579 
4728 

3385 
3579 
3766 
3945 
4116 
4281 
4440 
4594 
4742 

3404 
3598 
3784 
3962 
4133 
4298 
4456 
4609 
4757 

21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 

4914 
5052 
5185 
5315 
5441 
5563 
5682 
5798 
5911 

4928 
5065 
5198 
5328 
5453 
5575 
5694 
5809 
5922 

4942 
5079 
5211 
5340 
5465 
5587 
5705 
5821 
5933 

4955 
5092 
5224 
5353 
5478 
5599 
5717 
5832 
5944 

4969 
5105 
5237 
5366 
5490 
5611 
5729 
5843 
5955 

4983 
5119 
5250 
5378 
5502 
5623 
5740 
5855 
5966 

4997 
5132 
5263 
5391 
5515 
5635 
5752 
5866 
5977 

5011 
5145 
5276 
5403 
5527 
5647 
5763 
5877 
5988 

5024 
5159 
5289 
5416 
5539 
5658 
5775 
5888 
5999 

5038 
5172 
5302 
5428 
5551 
5670 
5786 
5899 
6010 

31 
32 
33 
34 
35 
36 
37 
38 
39 

40 

6021- 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

6128 
6232 
6335 
6435 
6532 
6628 
6721 
6812 
6902 

6138 
6243 
6345 
6444 
6542 
6637 
6730 
6821 
6911 

6149 
6253 
6355 
6454 
6551 
6646 
6739 
6830 
6920 

6160 
6263 
6365 
6464 
6561 
6656 
6749 
6839 
6928 

6170 
6274 
6375 
6474 
6571 
6665 
6758 
6848 
6937 

6180 
6284 
6385 
6484 
6580 
6675 
6767 
6857 
6946 

6191 
6294 
6395 
6493 
6590 
6684 
6776 
6866 
6955 

6201 
6304 
6405 
6503 
6599 
6693 
6785 
6875 
6964 

6212 
6314 
6415 
6513 
6609 
6702 
6794 
6884 
6972 

6222 
6325 
6425 
6522 
6618 
6712 
6803 
6893 
6981 

41 
42 
43 
44 
45 
46 
47 
48 
49 

50 

6990 

6998 

7007 

7016' 

7024 

7033 

7042 

7050 

7059 

7067 

50 

LOGARITHMS  OP  NUMBERS. 


169 


Four-place  Logarithms  of  Numbers  to  1000. 

For  six-place  logarithms  of  numbers  to  10,000,  see  pp.  137  to  164. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

No. 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

50 

51 
52 
53 

7076 
7160 

7243 

7084 
7168 
7251 

7093 
7177 
7259 

7101 

7185 
7267 

7110 
7193 
7275 

7118 
7202 
7284 

7126 
7210 
7292 

7135 
7218 
7300 

7143 
7226 
7308 

7152 
7235 
7316 

51 

52 
53 

54 
55 
56 

7324 
7404 
7482 

7332 
7412 
7490 

7340 
7419 
7497 

7348 
7427 
7505 

7356 
7435 
7513 

7364 
7443 
7520 

7372 
7451 
7528 

7380 
7459 
7536 

7388 
7466 
7543 

7396 
7474 
7551 

54 
55 
56 

57 
58 
59 

7559 
7634 
7709 

7566 
7642 
7716 

7574 
7649 
7723 

7582 
7657 
7731 

7589 
7664 
7738 

7597 
7672 

7745 

7604 
7679 
7752 

7612 
7686 
7760 

7619 
7694 
7767 

7627 
7701 
7774 

57 
58 
59 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

60 

61 
62 
63 

7853 
7924 
7993 

7860 
7931 
8000 

7868 
7938 
8007 

7875 
7945 
8014 

7882 
7952 
8021 

7889 
7959 
8028 

7896 
7966 
8035 

7903 
7973 
8041 

7910 
7980 
8048 

7917 
7987 
8055 

61 
62 
63 

64 
65 
66 

8062 
8129 
8195 

8069- 
8136 
8202 

8075 
8142 
8209 

8082 
8149 
8215 

8089 
8156 
8222 

8096 
8162 
8228 

8102 
8169 
8235 

8109 
8176 
8241 

8116 
8182 
8248 

8122 
8189 
8254 

64 
65 
66 

67 
68 
69 

8261 
8325 
8388 

8267 
8331 
8395 

8274 
8338 
8401 

8280 
8344 
8407 

8287 
8351 
8414 

8293 
8357 
8420 

8299 
8363 
8426 

8306 
8370 
8432 

8312 
8376 
8439 

8319 

8382 
8445 

67 
68 
69 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

70 

71 
72 
73 

8513 
8573 
8633 

8519 
8579 
8639 

8525 
8585 
8645 

8531 
8591 
8651 

8537 
8597 
8657 

8543 
8603 
8663 

8549 
8609 
8669 

8555 
8615 
8675 

8561 
8621 
8681 

8567 
8627 
8686 

71 
72 
73 

74 
75 
76 

8692 
8751 
8808 

8698 
8756 
8814 

8704 
8762 
8820 

8710 
8768 
8825 

8716 
8774 
8831 

8722 
8779 
8837 

8727 
8785 
8842 

8733 
8791 
8848 

8739 
8797 
8854 

8745 
8802 
8859 

74 
75 
76 

77 
78 
79 

8865 
8921 
8976 

8871 
8927 
8982 

8876 
8932 
8987 

8882 
8938 
8993 

8887 
8943 
8998 

8893 
8949 
9004 

8899 
8954 
9009 

8904 
8960 
9015 

8910 
8965 
9020 

8915 
8971 
9025 

77 
78 
79 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

80 

81 
82 
83 

9085 
9138 
9191 

9090 
9143 
9196 

9096 
9149 
9201 

9101 
9154 
9206 

9106 
9159 
9212 

9112 
9165 
9217 

9117 
9170 
9222 

9122 
9175 
9227 

9128 
9180 
9232 

9133 
9186 
9238 

81 
82 
83 

84 
85 
86 

9243 
9294 
9345 

9248 
9299 
9350 

9253 
9304 
9355 

9258 
9309 
9360 

9263 
9315 
9365 

9269 
9320 
9370 

9274 
9325 
9375 

9279 
9330 
9380 

9284 
9335 
9385 

9289 
9340 
9390 

84 
85 
86 

87 
88 
89 

9395 
9445 
9494 

9400 
9450 
9499 

9405 
9455 
9504 

9410 
9460 
9509 

9415 
9465 
9513 

9420 
9469 
9518 

9425 
9474 
9523 

9430 
9479 
9528 

9435 
9484 
9533 

9440 
9489 
9538 

87 
88 
89 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

90 

91 
92 
93 

9590 
9638 
9685 

9595 
9643 
9689 

9600 
9647 
9694 

9605 
9652 
9699 

9609 
9657 
9703 

9614 
9661 
9708 

9619 
9666 
9713 

9624 
9671 
9717 

9628 
9675 
9722 

9633 
9680 
9727 

91 
92 
93 

94 
95 
96 

9731 
9777 
9823 

9736 
9782 
9827 

9741 
9786 
9832 

9745 
9791 
9836 

9750 
9795 
9841 

9754 
9800 
9845 

9759 
9805 
9850 

9764 
9809 
9854 

9768 
9814 
9859 

9773 
9818 
9863 

94 
95 
96 

97 
98 
99 

9868 
9912 
9956 

9872 
9917 
9961 

9877 
9921 
9965 

9881 
9926 
9969 

9886 
9930 
9974 

9890 
9934 
9978 

9894 
9939 
9983 

9899 
9943 
9987 

9903 
9948 
9991 

9908 
9952 
9996 

97 
98 
99 

100 

0000 

0004 

0009 

0013 

0017 

0022 

0026 

0030 

0035 

0039 

100 

170          NATURAL   TRIGONOMETRICAL   FUNCTIONS. 


NATURAL  TRIGONOMETRICAL,  FUNCTIONS. 


• 

M. 

Sine. 

Co- 
Vers. 

Cosec. 

Tang. 

Co  tan. 

Se- 
cant. 

Ver. 

Sin. 

Cosine. 

o 

0 

.00000 

1  .0000 

Infinite 

.00000 

Infinite 

.0000 

.00000 

1.0000 

1)0 

^ 

15 

.00436 

.99564 

229.18 

.00436 

229.18 

.0000 

.00001 

.99999 

45 

30 

.00873 

.99127 

114.59 

.00873 

114.59 

.0000 

.00004 

.99996 

30 

45 

.01309 

.98691 

76.397- 

.01309 

76.390 

.0001 

.00009 

.99991 

15 

1 

0 

.01745 

.98255 

57.299 

.01745 

57.290 

.0001 

.00015 

.99985 

89 

0 

15 

.02181 

.97819 

45.840. 

.02182 

45.829 

.0002 

.00024 

.99976 

45 

30 

.02618 

.97382 

38.202 

.02618 

38.188 

.0003 

.00034 

.99966 

30 

45 

.03054 

.96946 

32.746 

.03055 

32.730 

.0005 

.00047 

.99953 

15 

2 

0 

.03490 

.96510 

28.654 

.03492 

28.636 

.0006 

.00061 

.99939 

88 

0 

15 

.03926 

.96074 

25.471 

.03929 

25.452 

.0008 

.00077 

.99923 

45 

30 

.G4362 

.95638 

22.926 

.04366 

22.904 

.0009 

.00095 

.99905 

30 

45 

.04798 

.95202 

20.843 

.04803 

20.819 

.0011 

.00115 

.99885 

15 

3 

0 

.05234 

.94766 

19.107 

.05241 

19.081 

.0014 

.00137 

.99863 

87 

0 

15 

.05669 

.9433  1 

17.639 

.05678 

17.611 

.0016 

.00161 

.99839 

45 

30 

.06105 

.93895 

16.380 

.06116 

16.350 

.0019 

.00187 

.9981.3 

30 

45 

.06540 

.93460 

15.290 

.06554 

15.257 

.0021 

.00214 

.99786 

15 

4 

0 

.06976 

.93024 

14.336 

.06993 

14.301 

.0024 

.00244 

.99756 

80 

0 

15 

.07411 

.92589 

13.494 

.07431 

13.457 

.0028 

.00275 

.99725 

45 

30 

.07846 

.92154 

12.745 

.07870 

12.706 

.0031 

.00308 

.99692 

30 

45 

.08281 

.91719 

12.076 

.08309 

12.035 

.0034 

.00343 

.99656 

15 

5 

0 

.08716 

.91284 

11.474 

.08749 

11.430 

.0038 

.00381 

.99619 

85 

0 

15 

.09150 

.90850 

10.929 

.09189 

10.883 

.0042 

.00420 

.99580 

45 

30 

.09585 

.90415 

10.433 

.09629 

10.385 

.0046 

.00460 

.99540 

30 

45 

.10019 

.89981 

9.9812 

.10069 

9.9310 

.0051 

.00503 

.99497 

15 

6 

0 

.10453 

.89547 

9.5668 

.10510 

9.5144 

.0055 

.00548 

.99452 

84 

0 

15 

.10887 

.89113 

9.1855 

.10952 

9.1309 

.0060 

.00594 

.99406 

45 

30 

.11320 

.88680 

8.8337 

.11393 

8.7769 

.0065 

.00643 

.99357 

30 

45 

.11754 

.88246 

8.5079 

.11836 

8.4490 

.0070 

.00693 

.99307 

15 

7 

0 

.12187 

.87813 

8.2055 

.12278 

8.1443 

.0075 

.00745 

.99255 

83 

0 

15 

.12620 

.87380 

7.9240 

.12722 

7.8606 

.0081 

.00800 

.99200 

45 

30 

.13053 

.86947 

7.6613 

.13165 

7.5958 

.0086 

.00856 

.99144 

30 

45 

.13485 

.86515 

7.4156 

.13609 

7.3479 

.0092 

.00913 

.99086 

15 

8 

0 

.13917 

.86083 

7.1853 

.14054 

7.1154 

.0098 

.00973 

.99027 

82 

0 

15 

.14349 

.85651 

6.9690 

.14499 

6.8969 

.0105 

.01035 

.98965 

45 

30 

.14781 

.85219 

6.7655 

.14945 

6.6912 

.0111 

.01098 

.98902 

30 

45 

.15212 

.84788 

6.5736 

.15391 

6.4971 

.0118 

.01164 

.98836 

15 

9 

0 

.15643 

.84357 

6.3924 

.15838 

6.3138 

.0125 

.01231 

.98769 

81 

0 

15 

.16074 

.83926 

6.2211 

.16286 

6.1402 

.0132 

.01300 

.98700 

45 

30 

.16505 

.83495 

6.0589 

.16734 

5.9758 

.0139 

.01371 

.98629 

30 

45 

.16935 

.83065 

5.9049 

.17183 

'5.8197 

.0147 

.01444 

.98556 

15 

10 

0 

.17365 

.82635 

5.7588 

.17633 

5.6713 

.0154 

.01519 

.98481 

80 

0 

15 

.17794 

.82206 

5.6198 

.18083 

5.5301 

.0162 

.01596 

.98404 

45 

30 

.18224 

.81776 

5.4874 

.18534 

5.3955 

.0170 

.01675 

.98325 

30 

45 

.18652 

.81348 

5.3612 

.18986 

5.2672 

.0179 

.01755 

.98245 

15 

11 

0 

.19081 

.80919 

5.2408 

.19438 

5.1446 

.0187 

.01837 

.98163 

79 

0 

15 

.19509 

.80491 

5.1258 

.19891 

5.0273 

.0196 

,01921 

.98079 

45 

30 

.19937 

.80063 

5.0158 

.20345 

4.9152 

.0205 

.02008 

.97992 

30 

45 

.20364 

.79636 

4.9106 

.20800 

4.8077 

.0214 

.02095 

.97905 

15 

13 

0 

.20791 

.79209 

4.8097 

.21256 

4.7046 

.0223 

.02185 

.97815 

78 

0 

15 

.21218 

.78782 

4.7130 

21712 

4.6057 

.0233 

.02277 

.97723 

45 

30 

.21644 

.78356 

4.6202 

.22169 

4.5107 

.0243 

.02370 

.97630 

30 

45 

.22070 

.77930 

4.5311 

.22628 

44194 

.0253 

.02466 

.97534 

15 

13 

0 

.22495 

.77505 

4.4454 

.23087 

4.3315 

.0263 

.02563 

.97437 

77 

0 

15 

.22920 

.77080 

4.3630 

.23547 

4.2468 

.0273 

.02662 

.97338 

45 

30 

.23345 

.76655 

4.2837 

.24008 

4.1653 

.0284 

.02763 

.97237 

30 

45 

.23769 

.76231 

4.2072 

.24470 

4.0867 

.0295 

.02866 

.97134 

15 

14 

0 

24192 

.75808 

4.1336 

.24933 

4.0108 

.0306 

.02970 

.97030 

76 

0 

15 

.24615 

.75385 

4.0625 

25397 

3.9375 

.0317 

.03077 

.96923 

45 

30 

.25038 

.74962 

3.9939 

.25862 

3.8667 

.0329 

.03185 

.96815 

30 

45 

.25460 

.74540 

3.9277 

.26328 

3.7983 

.0341 

.03295 

.96705 

15 

15 

0 

.25882 

.74118 

3.8637 

.26795 

3.7320 

1.0353 

.03407 

.96593 

!•> 

0 

Co- 
sine. 

Ver. 

Sin. 

Secant. 

Cotan 

Tang. 

Cosec. 

Co- 
Vers. 

Sine. 

0 

M; 

From  75°  to  90°  read  from  bottom  of  table  upwards. 


NATURAL  TRIGONOMETRICAL   FUNCTIONS. 


171 


• 

M. 

Sine. 

Co- 
Vers. 

Cosec 

Tang 

Cotan 

Secant 

Ver. 

Sin. 

Cosine 

I 

15~ 

~0~ 

.25882 

.74118 

3.8637 

.26795 

3.7320 

1.0353 

.03407 

.96593 

75 

~ 

15 

.26303 

.73697 

3.8018 

.27263 

3.6680 

1.0365 

.03521 

.96479 

45 

30 

.26724 

.73276 

3.7420 

.27732 

3.6059 

1.0377 

.03637 

.96363 

30 

45 

.27144 

.72856 

3.6840 

.28203 

3.5457 

1.C390 

.03754 

.96246 

15 

16 

0 

.27564 

.72436 

3.6280 

.28674 

3.4874 

1  .0403 

.03874 

.96126 

74 

0 

15 

.27983 

.72017 

3.5736 

.29147 

3.4308 

1.0416 

.03995 

.96005 

45 

30 

.28402 

.71598 

3.5209 

.29621 

3.3759 

1.0429 

.04118 

.95882 

30 

45 

.28820 

.71180 

3.4699 

.30096 

3.3226 

1  .0443 

.04243 

.95757 

15 

17 

0 

.29237 

.70763 

3.4203 

.30573 

3.2709 

1.0457 

.04370 

.95630 

73 

0 

15 

.29654 

.70346 

3.3722 

.31051 

3.2205 

1.0471 

.04498 

.95502 

45 

30 

.30070 

.69929 

3.3255 

.31530 

3.1716 

1  .0485 

.04628 

.95372 

30 

45 

.30486 

69514 

3.2801 

.32010 

3.1240 

1.0500 

.04760 

.95240 

15 

18 

0 

.30902 

69098 

3.2361 

.32492 

3.0777 

1.0515 

.04894 

.95106 

72 

0 

15 

31316 

68684 

3.1932 

.32975 

3.0326 

1.0530 

.05030 

.94970 

45 

30 

31730 

.68270 

3.1515 

.33459 

2.9887 

1.0545 

.05168 

.94832 

30 

45 

32144 

67856 

3.1110 

.33945 

2.9459 

1  .0560 

.05307 

.94693 

15 

19 

0 

32557 

67443 

3.0715 

34433 

2.9042 

1.0576 

.05448 

.94552 

71 

0 

15 

32969 

67031 

3.0331 

.34921 

2.8636 

1  .0592 

.05591 

.94409 

45 

30 

33381 

66619 

2.9957 

35412 

2.8239 

1  .0608 

.05736 

.94264 

30 

45 

33792 

66208 

2.9593 

35904 

2.7852 

1  .0625 

.05882 

.94118 

!5 

20 

0 

34202 

65798 

2.9238 

36397 

2.7475 

1  .0642 

.0603  1 

.93969 

70 

0 

15 

34612 

65388 

2.8892 

36892 

2.7106 

1  .0659 

.06181 

.93819 

45 

30 

35021 

64979 

2.8554 

37388 

2.6746 

1.0676 

.06333 

.93667 

30 

45 

35429 

64571 

2.8225 

37887 

2.6395 

1  .0694 

,.06486 

.93514 

1) 

21 

0 

35837 

64163 

2.7904 

38386 

2.6051 

1.0711 

.06642 

.93358 

69 

0 

15 

36244 

63756 

2.7591 

38888 

2.5715 

1.0729 

.06799 

.93201 

45 

30 

36650 

63350 

2.7285 

39391 

2.5386 

1.0748 

.06958 

.93042 

30 

45 

37056 

62944 

2.6986 

39896 

2.5065 

1.0766 

.07119 

.92881 

13 

22 

0 

37461 

62539 

2.6695 

40403 

2.4751 

1.0785 

.07282 

.92718 

68 

0 

15 

37865 

62135 

2.6410 

40911 

2.4443 

1  .0804 

.07446 

.92554 

45 

30 

38268 

61732 

2.6131 

41421 

2.4142 

1  .0824 

.07612 

.92388 

30 

45 

38671 

61329 

2.5859 

41933 

2.3847 

1  .0844 

.07780 

.92220 

15 

23 

0 

39073 

60927 

2.5593 

42447 

2.3559 

1  .0864 

.07950 

.92050 

67 

0 

15 

39474 

60526 

2.5333 

42963 

2.3276 

1  .0884 

.08121 

.91879 

45 

30 

39875 

60125 

2.5078 

43481 

2.2998 

1  .0904 

.08294 

.91706 

30 

45 

40275 

59725 

2.4829 

44001 

2.2727 

1  .0925 

.08469 

.91531 

15 

24 

0 

40674 

59326 

2.4586 

44523 

2.2460 

1.0946 

.08645 

.91355 

66 

0 

15 

41072 

58928 

2.4348 

45047 

2.2199 

1.0968 

.08824 

.91176 

45 

30 

41469 

58531 

2.4114 

45573 

2.1943 

1  .0989 

.09004 

.90996 

30 

45 

41866 

58134 

2.3886 

46101 

2.1692 

1.1011 

.09186 

.90814 

15 

25 

0 

42262 

57738 

2.3662 

46631 

2.1445 

1.1034 

.09369 

.90631 

65 

0 

15 

42657 

57343 

2.3443 

47163 

2.1203 

1.1056 

.09554 

.90446 

45 

30 

43051 

56949 

2.3228 

47697 

2.0965 

1.1079 

.09741 

.90259 

30 

45 

43445 

56555 

2.3018 

48234 

2.0732 

1.1102 

.09930 

.90070 

15 

26 

0 

43837 

56163 

2.2812 

48773 

2.0503 

1.1126 

.10121 

.89879 

64 

0 

15 

44229 

55771 

2.2610 

49314 

2.0278 

1.115.0 

.10313 

.89687 

45 

30 

44620 

55380 

2.2412 

49858 

2.0057 

1.1174 

.10507 

.89493 

30 

45 

45010 

54990 

2.2217 

50404 

1  .9840 

1.1198 

.10702 

.89298 

15 

27 

0 

45399 

54601 

2.2027 

50952 

1  .9626 

1.1223 

.10899 

.89101 

63 

0 

15 

45787 

54213 

2.1840 

51503 

1.9416 

1.1248 

.11098 

88902 

45 

30 

46175 

53825 

2.1657 

52057 

1.9210 

1.1274 

.11299 

.88701 

30 

45 

46561 

53439 

2.1477 

52612 

1  .9007 

1.1300 

.11501 

.88499 

15 

23 

0 

46947 

53053 

2.1300 

53171 

1  .8807 

1.1326 

.11705 

.88295 

62 

0 

15 

47332 

52668 

2.1127 

53732 

1.8611 

1.1352 

.11911 

.88089 

45 

30 

47716 

52284 

2.0957 

54295 

1.8418 

1.1379 

.12118 

.87882 

30 

45 

48099 

51901 

2.0790 

54862 

1.8228 

1.1406 

.12327 

.87673 

15 

29 

0 

48481 

51519 

2.0627 

55431 

1  .8040 

1.1433 

.12538 

.87462 

61 

0 

15 

48862 

51138 

2.0466 

56003 

1.7856 

1.1461 

.12750 

.87250 

45 

30 

49242 

50758 

2.0308 

56577 

1.7675 

1.1490 

.12964 

.87036 

30 

45 

49622 

.50378 

2.0152 

57155 

1.7496 

1.1518 

.13180 

.86820 

15 

30 

0 

50000 

.50000 

2.0000 

57735 

1.7320 

1.1547 

.13397 

.86603 

60 

_0 

Co- 
sine. 

Ver. 
Sin. 

Se- 
cant. 

Co  tan. 

Tang.. 

Cosec. 

Co- 
Vers. 

Sine. 

o 

M. 

From  60°  to  75°  read  from  bottom  of  table  upwards. 


172         NATUKAL   TRIGONOMETRICAL    FUNCTIONS. 


o 

M. 

Sine. 

Co- 
Vers. 

Cosec. 

Tang. 

Co  tan. 

Secant. 

Ver. 
Sin. 

Cosine 

80~ 

0 

.50000 

.50000 

2.0000 

.57735 

.7320 

.1547 

.13397 

.86603 

60 

0 

15 

.50377 

.49623 

.9850 

.58318 

.7147 

.1576 

.13616 

.86384 

45 

30 

.50754 

.49246 

.9703 

.58904 

.6977 

.1606 

.13837 

.86163 

30 

45 

.51129 

.48871 

.9558 

.59494 

.6808 

.1636 

.14059 

.85941 

15 

31 

0 

.51504 

.48496 

.9416 

.60086 

.6643 

.1666 

.14283 

.85717 

59 

0 

15 

.51877 

.48123 

.9276 

.60681 

.6479 

.1697 

.14509 

.85491 

45 

30 

.52250 

.47750 

.9139 

.61280 

.6319 

.1728 

.14736 

.85264 

30 

45 

.52621 

.47379 

.9004 

.61882 

.6160 

.1760 

.14965 

.85035 

15 

33 

0 

.52992 

.47008 

.8871 

.62487 

.6003 

.1792 

.15195 

.84805 

58 

0 

15 

.53361 

.46639 

.8740 

.63095 

.5849 

.1824 

.15427 

.84573 

45 

30 

.53730 

.46270 

.8612 

.63707 

.5697. 

.1857 

.15661 

.84339 

30 

45 

.54097 

.45903 

.8485 

.64322 

.5547 

.1890 

.15896 

.84104 

15 

33 

0 

.54464 

.45536 

.8361 

.64941 

5399 

.1924 

.16133 

.83867 

67 

0 

15 

.54829 

.45171 

.8238 

.65563 

.5253 

.1958 

.16371 

.83629 

45 

30 

.55194 

.44806 

.8118 

.66188 

.5108 

.1992 

.16611 

.83389 

30 

45 

.55557 

.44443 

.7999 

.66818 

.4966 

.2027 

.16853 

.83147 

15 

34 

0 

.55919 

.44081 

.7883 

.67451 

.4826 

.2062 

.17096 

.82904 

56 

0 

15 

.56280 

.43720 

.7768 

.68087 

.4687 

.2098 

.17341 

.82659 

45 

30 

.56641 

.43359 

.7655 

.68728 

.4550 

.2134 

.17587 

.82413 

30 

45 

.57000 

.43000 

.7544 

.69372 

.4415 

.2171 

.17835 

.82165 

15 

35 

0 

.57358 

.42642 

.7434 

.70021 

.4281 

.2208 

.18085 

.81915 

55 

0 

15 

.57715 

.42285 

.7327 

.70673 

.4150 

.2245 

.18336 

.81664 

45 

30 

.58070 

.41930 

.7220 

.71329 

.4019 

.2283 

.18588 

.81412 

30 

45 

.58425 

.41575 

.7116 

.71990 

.3891 

.2322 

.18843 

.81157 

15 

36 

0 

.58779 

.41221 

.7013 

.72654 

.3764 

.2361 

.19098 

.80902 

54 

0 

15 

.59131 

.40869 

.6912 

.73323 

.3638 

.2400 

.19356 

.80644 

45 

30 

.59482 

.40518 

.6812 

.73996 

.3514 

.2440 

.19614 

.80386 

30 

45 

.59832 

.40168 

.6713 

.74673 

.3392 

.2480 

.19875 

.80125 

15 

37 

0 

.60181 

.39819 

.6616 

.75355 

.3270 

.2521 

.20136 

.79864 

53 

0 

15 

.60529 

.39471 

.6521 

.76042 

.3151 

.2563 

.20400 

.79600 

45 

30 

.60876 

.39124 

.6427 

.76733 

.3032 

.2605 

.20665 

.79335 

30 

45 

.61222 

.38778 

.6334 

.77428 

.2915 

.2647 

.20931 

.79069 

15 

38 

0 

.61566 

.38434 

.6243 

.78129 

.2799 

.2690 

.21199 

.78801 

52 

0 

15 

.61909 

.38091 

.6153 

78834 

.2685 

.2734 

.21468 

.78532 

45 

30 

.62251 

.37749 

.6064 

.79543 

.2572 

.2778 

.21739 

.78261 

30 

45 

.62592 

.37408 

5976 

.80258 

.2460 

2822 

.22012 

.77988 

15 

39 

0 

.62932 

.37068 

.5890 

.80978 

.2349 

.2868 

.22285 

.77715 

51 

0 

15 

.63271 

.36729 

.5805 

.81703 

.2239 

.2913 

.22561 

.77439 

45 

30 

.63608 

.36392 

.5721 

.82434 

.2131 

.2960 

.22833 

.77162 

30 

45 

.63944 

.36056 

.5639 

.83169 

.2024 

.3007 

.23116 

.76884 

15 

40 

0 

.64279 

.35721 

.5557 

.83910 

.1918 

.3054 

.23396 

.76604 

50 

0 

15 

.64612 

.35388 

.5477 

.84656 

.1812 

.3102 

.23677 

.76323 

45 

30 

.64945 

.35055 

.5398 

.85408 

.1708 

.3151 

.23959 

.76041 

30 

45 

.65276 

.34724 

.5320 

.86165 

.1606 

.3200 

.24244 

.75756 

15 

41 

0 

.65606 

.34394 

.5242 

.86929 

.1504 

.3250 

.24529 

.75471 

49 

0 

15 

.65935 

.34065 

.5166 

.87698 

.1403 

.3301 

.24816 

.75184 

45 

30 

.66262 

.33738 

.5092 

.88472 

.1303 

.3352 

.25104 

.74896 

30 

45 

.66588 

.33412 

.5018 

.89253 

.1204 

.3404 

.25394 

.74606 

15 

42 

0 

.66913 

.33087 

.4945 

.90040 

.1106 

.3456 

.25686 

.74314 

48 

0 

15 

.67237 

.32763 

.4873 

.90834 

.1009 

.3509 

.25978 

.74022 

45 

30 

.67559 

.32441 

.4802 

.91633 

.0913 

.3563 

.26272 

.73728 

30 

45 

.67880 

.32120 

.4732 

.92439 

.0818 

.3618 

.26568 

.73432 

15 

43 

0 

.68200 

.31800 

.4663 

.93251 

.0724 

.3673 

.26865 

.73135 

47 

0 

15 

.68518 

.31482 

.4595 

.94071 

.0630 

.3729 

.27163 

.72837 

45 

30 

.68835 

.31165 

.4527 

.94896 

.0538 

.3786 

.27463 

.72537 

30 

45 

.69151 

.30849 

.4461 

.95729 

.0446 

.3843 

.27764 

.72236 

15 

44 

0 

.69466 

.30534 

.4396 

.96569 

.0355 

.3902 

.28066 

.71934 

46 

0 

15 

.69779 

.30221 

.4331 

.97416 

.0265 

.3961 

.28370 

.71630 

45 

30 

.70091 

.29909 

.4267 

.98270 

.0176 

.4020 

.28675 

.71325 

30 

45 

.70401 

.29599 

.4204 

.99131 

.0088 

.4081 

.28981 

.71019 

15 

45 

0 

.70711 

.29289 

.4142 

1  .0000 

.0000 

.4142 

.29289 

.70711 

45 

0 

Cosine 

Ver. 
Sin. 

Se- 
cant. 

Cotan. 

Tang. 

Cosec. 

Co- 
Vers. 

Sine. 

• 

M. 

From  45°  to  60°  read  from  bottom  of  table  upwards. 


SPECIFIC    GRAVITY. 


173 


MATERIALS. 

THE   CHEMICAL  ELEMENTS. 

Common  Elements  (42). 


•11 

«£ 

12 

«£ 

V 

OrC 

Name. 

|-s 

IJ 

Name. 

1^ 

fl 

Name. 

J't? 

Al 

Sb 

Aluminum 
Antimony 

27.1 
120.2 

F 
Au 

Fluorine 
Gold 

19. 
197.2 

Pd 
P 

Palladium 
Phosphorus 

106.7 
31. 

As 

Arsenic 

75.0 

H 

Hydrogen 

1.01 

Pt 

Platinum 

195.2 

Ba 

Barium 

137.4 

I 

Iodine 

126.9 

K 

Potassium 

39.1 

Bi 

Bismuth 

208.0 

Ir 

Iridium 

193.1 

Si 

Silicon 

28.3 

B 

Boron 

11.0 

Fe 

Iron 

55.84 

Ag 

Silver 

107.9 

Br 

Bromine 

79.9 

Pb 

Lead 

207.2 

Na 

Sodium 

23. 

Cd 

Cadmium 

112.4 

Li 

Lithium 

6.94 

Sr 

Strontium 

87.6 

Ca 

Calcium 

40.1 

Mg 

Magnesium 

24.34 

S 

Sulphur 

32.1 

C 

Carbon 

12. 

Mn 

Manganese 

54.9 

Sn 

Tin 

119. 

Cl 

Chlorine     • 

35.5 

Hg 

Mercury 

200.6 

Ti 

Titanium 

48.1 

Cr 

Chromium 

52.0 

Ni 

Nickel 

58.7 

W 

Tungsten 

184.0 

Co 

Cobalt 

59. 

N 

Nitrogen 

14.01 

Va 

Vanadium 

51.0 

Cu 

Copper 

63.6 

0 

Oxygen 

16. 

Zn     1 

Zinc 

65.4 

The  atomic  weights  of  many  of  the  elements  vary  in  the  decimal 
place  as  given  by  different  authorities.  The  above  are  the  most  recent 
values  referred  to  O  =  16  and  H  =  1.008.  When  H  is  taken  as  1, 
O  =  15.879,  and  the  other  figures  are  diminished  proportionately. 

Rare  Elements  (37). 

Beryllium,  Be.  Indium,  In.  Ruthenium,  Ru.  Thallium,  Tl. 

Caesium,  Cs.  Lanthanum,  La.  Samarium,  Sm.  Thorium,  Th. 

Cerium,  Ce.  Molybdenum,  Mo.  Scandium,  Sc.  Uranium,  U. 

Erbium,  Er.  Niobium,  Nb.  Selenium,  Se.  Ytterbium,  Yr. 

Gallium,  Ga.  Osmium,  Os.  Tantalum,  Ta.  Yttrium,  Y. 

Germanium,  Ge.  Rhodium,  R.  Tellurium,  Te.  Zirconium,  Zr. 

Glucinum,  G.  Rubidium,  Rb.  Terbium,  Tb. 

Elements  recently  discovered  (1895-1900):  Argon,  A,  39.9;  Krypton 
Kr,  81.8;  Neon,  Ne,  20.0;  Xenon,  X,  128.0;  constituents  of  the  atmos- 
phere, which  contains  about  1  per  cent  by  volume  of  Argon,  and  very 
small  quantities  of  the  others.  Helium,  He,  4.0;  Radium,  Ra,  225.0; 
Gadolinium,  Gd,  156.0;  Neodymium.  Nd,  143.6;  Praesodymium,  Pr, 
110.5;  Thulium,  Tm,  171.0. 


SPECIFIC  GRAVITY. 

The  specific  gravity  of  a  substance  is  its  weight  as  compared  with  the 
weight  of  an  equal  bulk  of  pure  water.  In  the  metric  system  it  is  the 
weight  in  grammes  per  cubic  centimeter. 

To  find  the  specific  gravity  of  a  substance; 

W  =  weight  of  body  in  air ;  w  =  weight  of  body  submerged  in  water. 


Specific  gravity  = 


W 


W  -w' 


If  the  substance  be  lighter  than  the  water,  sink  it  by  means  of  a 
heavier  substance,  and  deduct  the  weight  of  the  heavier  substance. 

Specific  gravity  determinations  are  usually  referred  to  the  standard  of 
the  weight  of  water  at  62°  F.,  62.355  Ib.  per  cubic  foot.  Some  expert- 


174 


MATERIALS. 


menters  have  used  60°  F.  as  the  standard,  and  others  32°  and  39.1°  F. 
There  is  no  general  agreement. 

Given  sp.  gr.  referred  to  water  at  39.1°  F.,  to  reduce  it  to  the  standard 
of  62°  F.  multiply  it  by  1.00112. 

Given  sp.  gr.  referred  to  water  at  62°  F.,  to  find  weight  per  cubic  foot 
multiply  by  62.355.  Given  weight  per  cubic  foot,  to  find  sp.  gr.  multiply 
by  0.016037.  Given  sp.  gr.,  to  find  weight  per  cubic  inch  multiply  by 

Weight  and  Specific  Gravity  of  Metals. 


Specific  Gravity. 
Range  accord- 
ing to 
several 
Authorities. 

Specific  Grav- 
ity.    Approx. 
Mean  Value, 
used  in 
.    Calculation 
of  Weight. 

Weight 
per 
Cubic 
Foot, 
Ibs. 

Weight 
per 
Cubic 
Inch, 
Ibs. 

2.56     to     2.71 

2.67 

166.5 

00963 

Antimony  

6.66     to     6.86 

6  76 

421  6 

02439 

Bismuth  

9.74     to     9.90 

9.82 

612.4 

0.3544 

Brass:  Copper  +  Zinc-K 
80             20 
70             30L  . 
60             40 
50             50* 

Cadmium    .  . 

7.8       to     8.6 

8.52     to     8.96 
8.6       to     8.7 

{8.60 
8.40 
8.36 
8.20 

8.853 
865 

536.3 
523.8 
521.3 
511.4 

552. 
539 

0.3103 
0.3031 
0.3017 
0.2959 

0.3195 
03121 

Calcium  

1.58 

1.58 

98.5 

0.0570 

Ch  rom  i  um 

50 

5  0 

311  8 

0  1804 

Cobalt       

85       to     8  6 

8.55 

533  1 

0  3085 

19.245  to   19.361 

19.258 

1200.9 

06949 

Copper    .    .  . 

8.69     to     8  92 

8853 

552 

03195 

Iridium     

22.38     to  23. 

22.38 

1396 

08076 

Iron   Cast 

6  85     to     7  48 

7218 

450 

02604 

Iron   Wrought  

7.4       to     7.9 

7  70 

480 

02779 

Lead  

11.07     to   11.44 

11.38 

709.7 

04106 

Manganese  ... 

7.         to     8. 

8. 

499 

02887 

Magnesium.  .  , 

1  .69     to     1  .75 

1.75 

109. 

0.0641 

j  32° 
Mercury  <  60° 
1212° 
Nickel 

13.61 
13.58 
13.37     to  13.38 
8.279  to     8.93 

13.61 
13.58 
13.38 
8.8 

848.6 
846.8 
834.4 
548  7 

0.4908 
0.49  1  1 
0.4828 
03175 

Platinum  

20.33     to  22.07 

21.5 

1347.0 

07758 

0.865 

0.865 

53.9 

0.0312 

Silver  

10.474  to   10.511 

10.505 

655.1 

03791 

Sodium  

0.97 

0.97 

60.5 

0.0350 

Steel... 

7  69*  to     7.932t 

7.854 

4896 

02834 

Tin  

7.291  to     7.409 

7.350 

458.3 

0.2652 

Titanium  .    . 

5.3 

5.3 

330  5 

0  1913 

17.        to  17.6 

17.3 

1078.7 

0.6243 

Zinc.  .  . 

6.86     to     7.20 

7.00 

436.5 

0.2526 

*  Hard  and  burned. 

t  Very  pure  and  soft.     The  sp.  gr.  decreases  as  the  carbon  is  increased. 

In  the  first  column  of  figures  the  lowest  are  usually  those  of  cast  metals, 
which  are  more  or  less  porous;  the  highest  are  of  metals  finely  rolled  or 
drawn  into  wire. 

The  weight  of  1  cu.  cm.  of  mercury  at  0°  C.  is  13.59545  grams  (Thiessen). 
Taking  atmosphere  =  29.92  in.  of  mercury  at  32°  F.  =  14.6963  Ib.  per 
sq.  in.,  1  cu.  im  of  mercury  =  0.49117  Ib.  Taking  water  at  0.036085  Ib. 
per  cu.  in.  at  62°  F.,  the  specific  gravity  of  mercury  is  at  32°  F.  13.611. 


SPECIFIC   GKAVITY. 


175 


Specific  Gravity  of  Liquids  at  60°  F. 


A   'r\    AT   *•'   tV 

I  200 

Naphtha           

0.670  to  0.737 

"      Nitric 

1.54 

0.93 

"     Sulphuric 

1  849 

"     Olive        

0.92 

Alcohol  pure 

0.794 

"    Palm  

0.97 

"        95  per  cent  .  . 
"        50  per  cent 

0.816 
0.934 

'    Petroleum,  crude. 
"    Rape 

0.78  to   1.00 
0.92 

Ammonia   27  9  per  ct 

0.891 

*    Turpentine  

0.86 

Bromine 

2.97 

"    Whale  

0.92 

Carbon  disulphide 

1.26 

Tar    

1.0 

Ether  Sulphuric 

0  72 

Vinegar 

1.08 

Gasoline 

0  660  to  0.670 

Water  

1.0 

Kerosene.  . 

0.753  to  0.864 

Water,  Sea  ... 

1.026  to  1.03 

Compression  of  the  following  Fluids  under  a  Pressure  of  15  Ib. 
per  Square  Inch. 

Water     0.00004663    I    Ether 0.00006158 

Alcohol 0.0000216      |    Mercury 0.00000265 

The  Hydrometer. 

The  hydrometer  is  an  instrument  for  determining  the  density  of 
liquids.  It  is  usually  made  of  glass,  and  consists  of  three  parts:  (1) 
the  upper  part,  a  graduated  stem  or  fine  tube  of  uniform  diameter; 
(2)  a  bulb,  or  enlargement  of  the  tube,  containing  air,  and  (3)  a  small 
bulb  at  the  bottom,  containing  shot  or  mercury  which  causes  the  in- 
strument to  float  in  a  vertical  position.  The  graduations  are  figures 
representing  either  specific  gravities,  or  the  numbers  of  an  arbitrary  scale, 
as  in  Baume's,  Twaddell's,  Beck's,  and  other  hydrometers. 

There  is  a  tendency  to  discard  all  hydrometers  with  arbitrary  scales 
and  to  use  only  those  which  read  in  terms  of  the  specific  gravity 
directly. 

Baume's  Hydrometer  and  Specific  Gravities  Compared. 


5  Heavy  liquids,  Sp.  gr. 
l  Light    liquids,  Sp.  gr. 


145  -r  (145  -deg.  Be.) 
140  -r  (130  +  deg.  Be.) 


Degrees 
Baume* 

Liquids 
Heavier 
than 
Water, 
Sp.  Gr. 

Liquids 
Lighter 
than 
Water, 
Sp.  Gr. 

Degrees 
Baume* 

Liquids 
Heavier 
than 
Water, 
Sp.  Gr 

Liquids 
Lighter 
than 
Water, 
Sp.  Gr. 

Degrees 
Baume* 

Liquids 
Heavier 
than 
Water, 
Sp.  Gr. 

Liquids 

Htr 

Water, 
Sp.  Gr. 

00 

000 

190 

151 

0940 

380 

355 

0833 

1.0 

.007 

20.0 

.160 

0.933 

39.0 

.368 

0.828 

2.0 
3.0 

.014 
021 

21.0 
22.0 

.169 
.179 

0.927 
0.921 

40.0 
41  0 

.381 
394 

0.824 
0  819 

4.0 

.028 

23.0 

189 

0915 

42  0 

408 

0  814 

5.0 

.036 

24.0 

.198 

0.909 

44.0 

.436 

0805 

6.0 
7.0 

.043 
.051 

25.0 
26.0 

.208 
.219 

0.903 
0.897 

46.0 
48.0 

.465 
.495 

0.796 
0.787 

8.0 
9.0 

.058 
.066 

27.0 
28.0 

.229 
.239 

0.892 
0.886 

50.0 
52.0 

.526 
.559 

0.778 
0.769 

10.0 
11.0 
12.0 
13.0 
14.0 
15.0 
16.0 
17.0 

.074 
.082 
.090 
.099 
.107 
.115 
.124 
.133 

1.000 
0.993 
0.986 
0.979 
0.972 
0.966 
0.959 
0.952 

29.0 
30.0 
31.0 
32.0 
330 
34.0 
35.0 
36.0 

.250 
.261 
.272 
.283 
.295 
.306 
1.318 
1.330 

0.881 
0.875 
0.870 
0.864 
0,859 
0.854 
0.849 
0.843 

54.0 
56.0 
58.0 
60.0 
65.0 
70.0 
75.0 

.593 
.629 
.667 
.706 
.813 
.933 
2.071 

0761 
0.753 
0.745 
0.737 
0.718 
0.700 
0.683 

18.0 

142 

0946 

370 

1  343 

0838 

170 


MATERIALS. 


Specific  Gravity  and  Weight  of  Gases  at  Atmospheric  Pressure 
and  33°  F. 

(For  other  temperatures  and  pressures  see  Physical  Properties  of  Gases.) 


Density, 
Air  =>  1. 

Density, 
H  =  f. 

Grammes 
per  Litre. 

Lbs.  per 
Cu.  Ft. 

Cubic  Ft. 
per  Lb. 

Air  

1  .0000 

1  4.444 

.2931 

0.080728 

12388 

1.1052 

15.963 

.4291 

0.08921 

11  209 

Hydrogen,  H  

0.0692 

1  000 

0.0895 

0.00559 

1  78  93  1 

Nitrogen,  N  

0.9701 

14.012 

.2544 

0.07831 

12  770 

Carbon  monoxide,  CO  . 
Carbon  dioxide,  CO2  .  . 
Metha  ne,marsh-gas,  CtU 
Ethyl  ene  C2EU  

0.9671 
1.5197 
0.5530 
0.9674 

13.968 
21.950 
7.987 
13.973 

.2505 
.9650 
0.7150 
.2510 

0.07807 
0.12267 
C.04464 
0  07809 

12.810 
8.152 
22.429 
12  805 

08982 

12.973 

.1614 

0.07251 

13.792 

0.5889 

8.506 

0.7615 

0  04754 

21  036 

Water  vapor,  HtO  .  .  ,  . 
Sulphur  dioxide,  SO2  .  . 

0.6218 
2.213 

8.981 
31.965 

0.8041 
2.862 

0.05C20 
0.1787 

19.922 
5.597 

Specific  Gravity  and  Weight  of  Wood. 


Specific 
Gravity. 

rii 

Sl| 

QjOPn 
£ 

Specific 
Gravity, 

&4 
%as 

«•§! 

uoPk 

Alder  

Avge. 
0.56  to  0.80  0.68 
0.73  to  0.79  0.76 
0.60  to  0.84  0.72 
0.31  to  0.40  0.35 
0.62  to  0.85   0.73 
0.56  to  0.74  0.65 
0.91  to  1.33    1.12 
0.49  to  0.75   0.62 
0.61  to  0.72   0.66 
0.46  to  0.66  0.56 
0.24                0.24 
0.41  to  0.66  0.53 
0.76                0.76 
1.13  to  1.33    1.23 
0.55  to  0.78  0.61 
0.48  to  0.70  0.59 
0.84  to  1  .00  0.92 
0.59                 0.59 
0.36  to  0.41    0.38 
0.69  to  0.94  0.77 
0.76                0.76 

42 
47 
45  • 
22 
46 
41 
70 
39 
41 
35 
15 
33 
47 
76 
33 
37 
57 
37 
24 
48 
47 

Hornbeam.  . 
Juniper  .... 
Larch  
Lignum  vita? 
Linden     .  .  . 
Locust  
Mahogany.  . 
Maple  

Avge. 
0.76                0.76 
0.56                0.56 
0.56                0.56 
0.65  to  1.33    1.00 
0.604 
0.728 
0.56  to  1.06  0.81 
0.57  to  0.79  0.68 
0.56  to  0.90   0.73 
0.96  to  1.26    1.11 
0.69  to  0.86  0.77 
0.73  to  0.75   0.74 
0.35  to  0.55   0.45 
0.46  to  0.76   0.61 
0.38  to  0.58  0.48 
0.40  to  0.50   0.45 
0.59  to  0.62   0.60 
0.66  to  0.98   0.82 
0.50  to  0.67   0.58 
0.49  to  0.59   0.54 

47 
35 
35 
62 
37 
46 
51 
42 
46 
69 
48 
46 
28 
38 
30 
28 
37 
51 
36 
34 

Ash  

Bamboo  .... 
Beech  

Birch  
Box 

Cedar. 

Cherry  
Chestnut.  .  .  . 
Cork 

Mulberry.  .  . 
Oak,  Live  .  . 
Oak,  White. 
Oak,  Red  .  . 
Pine,   White 
"  Yellow 
Poplar  
Spruce  
Sycamore  .  . 
Teak  

Cypress  
Dogwood  .  .  . 
Ebony.    . 

Elm.  .  . 

Fir  

Gum 

Hackmatack 
Hemlock.  .  .  . 
Hickory  
Holly.      . 

Walnut  
Willow  

OP   THE   USEFUL  METALS. 


177 


Weight  and  Specific  Gravity  of  Stones,  Brick,  Cement,  etc. 
Water  =  1.00.) 


(Pure 


Lb.  per  Cu.  Ft. 

Sp.  Gr. 

Ashes  

43 

87 

1.39 

Brick,  Soft                         

100 

1  .6 

112 

1.79 

Hard        

125 

2.0 

"       Pressed  

135 

2.16 

"       Fire                         

140  to  150 

2.  24  to  2  4 

Sand-lime  

136 

2.18 

Brickwork  in  mortar 

100 

1  6 

"    cement  .  .  %  

112 

1.79 

Cement,  American,  natural 

28    to  3  2 

"         Portland 

3  .  05  to  3   15 

loose  

92 

"         in  barrel 

115 

Clay  

120  to  150 

'  1  .92  to  2.4 

Concrete 

120  to  155 

1   92  to  2  48 

Earth,  loose  .        ... 

72  to     80 

1  .  1  5  to  1   28 

rammed 

90  to  110 

1   44  to  1   76 

Emery  . 

250 

4. 

Glass 

1  56  to  1  72 

25    to  2  75 

flint  

180  to  196 

2.88  to  3   14 

Gneiss    1 

160  to  170 

2.  56  to  2.72 

Granite  f  
Gravel  

100  to  120 

1.6    to  1.92 

Gypsum 

130  to   150 

2  08  to  2  4 

Hornblende  

200  to  220 

3.2    to  3.  52 

Ice      . 

55  to     57 

0  88  to  0  92 

Lime,  quick,  in  bulk  

50  to     60 

0.8    to  0.96 

Limestone 

140  to   185 

2  30  to  2  90 

Magnesia,  Carbonate  

150 

2.4 

Marble 

160  to   180 

2  56  to  2  88 

Masonry,  dry  rubble  

140  to   160 

2.24  to  2.56 

"         dressed 

140  to   180 

2  24  to  2  88 

Mica  

175 

2.80 

Mortar 

90  to   100 

44  to  1   6 

Mud,  soft  flowing  

104  to   120 

.67  to  1  .92 

Pitch 

72 

15 

Plaster  of  Paris  

93  to   113 

.50  to  T.81 

Quartz  

165 

2.64 

Sand                  .    .                .... 

90  to   110 

44  to  1   76 

"     wet  

118  to   129 

.89  to  2.  07 

Sandstone  .... 

140  to   150 

2  24  to  2.4 

Slate  

170  to   180 

2.72  to  2.88 

Soapstone  ... 

166  to   175 

2  65  to  2.8 

Stone,  various  

135  to  200 

2.16to3.4 

"      crushed  

100 

Tile 

1  10  to   120 

1   76  to  1   92 

Trap  Rock  

1  70  to  200 

2.72  to  3.  4 

PROPERTIES   OF  THE  USEFUL  METALS. 

Aluminum,  AI.  —  Atomic  weight  27.1.  Specific  gravity  2.6  to  2.7. 
The  lightest  of  all  the  useful  metals  except  magnesium.  A  soft,  ductile, 
malleable  metal,  of  a  white  color,  approaching  silver,  but  with  a  bluish 
cast.  Very  non-corrosive.  Tenacity  about  one-third  that  of  wrought 
iron.  Formerly  a  rare  metal,  but  since  1890  its  production  and  use 
have  greatly  increased  on  account  of  the  discovery  of  cheap  processes 
for  reducing  it  from  the  ore.  Melts  at  1215°  F.  For  further  description 
see  Aluminum,  under  Strength  of  Materials,  page  380. 

Antimony  (Stibium),  Sb. — At.  wt.  120.2  Sp.  gr.  6.7  to  6.8.  A 
brittle  metal  of  a  bluish-white  color  and  highly  crystaline  or  laminated 
structure.  Melts  at  842°  F.  Heated  in  the  open  air  it  burns  with  a 


178  MATERIALS. 

bluish-white  flame.  Its  chief  use  is  for  the  manufacture  of  certain  alloys,  j 
as  type-metal  (antimony  1,  lead  4),  britannia  (antimony  1,  tin  9),  and  4 
various  anti-friction  metals  (see  Alloys).  Cubical  expansion  by  heat  3 
from  32°  to  212°  F.,  0.0070.  Specific  heat  0.050. 

Bismuth,  Bi.  —  At.  wt.  208.5.  Bismuth  is  of  a  peculiar  light  reddish  I 
color,  highly  crystalline,  and  so  brittle  that  it  can  readily  be  pulverized,  j 
It  melts  at  510°  F.,  and  boils  at  about  2300°  F.  Sp.  gr.  9.823  at  54°  F.,  ] 
and  10.055  just  above  the  melting-point.  Specific  heat  about  0.0301  at  j 
ordinary  temperatures.  Coefficient  of  cubical  expansion  from  32°  to  i 
212°,  0.0040.  Conductivity  for  heat  about  1/56  and  for  electricity  only  .] 
about  i/so  of  that  of  silver.  Its  tensile  strength  is  about  6400  IDS.  per  | 
square  inch.  Bismuth  expands  in  cooling,  and  Tribe  has  shown  that  } 
this  expansion  does  not  take  place  until  after  solidification.  Bismuth  is  \ 
the  most  diamagnetic  element  known,  a  sphere  of  it  being  repelled  by  a  ; 
strong  magnet. 

Cadmium,  Cd.  —  At.  wt.  112.4.     Sp.  gr.  8.6  to  8.7.     A  bluish-white 
metal,  lustrous,  with  a  fibrous  fracture.     Melts  below  500°  F.  and  vola- 
tilizes at  about  680°  F.     It  is  used  as  an  ingredient,  in  some  fusible  alloys 
with  lead,  tin,  and  bismuth.     Cubical  expansion  from  32°  to  212°  F.,  • 
0.0094. 

Copper,  Cu.  —  At.  wt.  63.6.     Sp.  gr.  8.81  to  8.95.     Fuses  at  about  ; 
1930^  F.     Distinguished  from  all  other  metals  by  its  reddish  color.     Very 
ductile  and  malleable,  and  its  tenacity  is  next  to  iron.     Tensile  strength 
20,000  to  30,000  Ibs.  per  square  inch.     Heat  conductivity  73.6%  of  that  i 
of  silver,  and  superior  to  that  of  other  metals.     Electric  conductivity 
equal  to  that  of  gold  and  silver.     Expansion  by  heat  from  32°  to  212°  F., 
0.0051  of  its  volume.     Specific  heat  0.093.     (See  Copper  under  Strength 
of  Materials;  also  Alloys.) 

Gold  (Aurum),  Au.  —  At.  wt.  197.2.  Sp.  gr.,  when  pure  and  pressed 
in  a  die,  19.34.  Melts  at  about  1915°  F.  The  most  malleable  and  duc- 
tile of  all  metals.  One  ounce  Troy  may  be  beaten  so  as  to  cover  160  sq. 
ft.  of  surface.  The  average  thickness  of  gold-leaf  is  1/282000  of  an  inch,  i 
or  100  sq.  ft.  per  ounce.  One  grain  may  be  drawn  into  a  wire  500  ft.  in 
length.  The  ductility  is  destroyed  by  the  presence  of  1/2000  part  of  lead, 
bismuth,  or  antimony.  Gold  is  hardened  by  the  addition  of  silver  or  of 
copper.  U.  S.  gold  coin  is  90  parts  gold  and  10  parts  alloy,  which  is 
chiefly  copper  with  a  little  silver.  By  jewelers  the  fineness  of  gold  is 
expressed  in  carats,  pure  gold  being  24  carats,  three-fourths  fine  18 
carats,  etc. 

Iridium,  Ir.  —  Iridium  is  one  of  the  rarer  metals.  It  has  a  white 
lustre,  resembling  that  of  steel;  its  hardness  is  about  equal  to  that  of  the 
ruby;  in  the  C9ld  it  is  quite  brittle,  but  at  white  heat  it  is  somewhat 
malleable.  It  is  one  of  the  heaviest  of  metals,  having  a  specific  gravity 
of  22.38.  It  is  extremely  infusible  and  almost  absolutely  inoxidizable. 

For  uses  of  iridium,  methods  of  manufacturing  it,  etc.,  see  paper  by 
W.  L.  Dudley  on  the  "Iridium  Industry,"  Trans.  A.  I.  M.  E.,  1884. 

Iron  (Ferrum),Fe.  —  At.  wt.  55.9.  Sp.  gr.:  Cast,  6.85  to  7.48;  Wrought, 
7.4  to  7.9.  Pure  iron  is  extremely  infusible,  its  melting  point  being  above 
3000°  F.,  but  its  fusibility  increases  with  the  addition  of  carbon,  cast 
iron  fusing  ab9ut  2500°  F.  Conductivity  for  heat  11.9,  and  for  electricity 
12  to  14.8,  silver  being  100.  Expansion  in  bulk  by  heat:  cast  iron 
0.0033,  and  wrought  iron  0.0035,  from  32°  to  212°  F.  Specific,  heat: 
cast  iron  0.1298,  wrought  iron  0.1138,  steel  0.1165.  Cast  iron  exposed 
to  continued  heat  becomes  permanently  expanded  1  1/2  to  3  per  cent  of  its 
length.  Grate-bars  should  therefore  be  allowed  about  4  per  cent  play. 
(For  other  properties  see  Iron  and  Steel  under  Strength  of  Materials.) 

Lead  (Plumbum),  Pb.  —  At.  wt  206.9.  Sp.  gr.  11.07  to  11.44  by  dif- 
ferent authorities.  Melts  at  about  625°  F.,  softens  and  becomes  pasty 
at  about  617°  F.  If  broken  by  a  sudden  blow  when  just  below  the 
melting-point  it  is  quite  brittle  and  the  fracture  appears  crystalline. 
Lead  is  very  malleable  and  ductile,  but  its  tenacity  is  such  that  it  can 
be  drawn  into  wire  with  great  difficulty.  Tensile  strength,  1600  to 
2400  Ibs.  per  square  inch.  Its  elasticity  is  very  low,  and  the  metal 
flows  under  very  slight  strain.  Lead  dissolves  to  some  extent  in  pure 
water,  but  water  containing  carbonates  or  sulphates  forms  over  vt  » 
film  of  insoluble  salt  which  prevents  further  action. 


PROPERTIES   OF  THE  USEFUL  METALS.  179 

Magnesium,  Mg.  —  At.  wt.  24.36.  Sp.  gr.  1.69  to  1.75.  Silver-white, 
brilliant,  malleable,  and  ductile.  It  is  one  of  the  lightest  of  metals, 
weighing  only  about  tvyo  thirds  as  much  as  aluminum.  In  the  form  of 
filings,  wire,  9r  thin  ribbons  it  is  highly  combustible,  burning  with  a 
light  of  dazzling  brilliancy,  useful  for  signal-lights  and  for  flash-lights 
for  photographers.  It  is  nearly  non-corrosive,  a  thin  film  of  carbonate 
of  magnesia  forming  on  exposure  to  damp  air,  which  protects  it  from 
further  corrosion.  It  may  be  alloyed  with  aluminum,  5  per  cent  Mg 
added  to  Al  giving  about  as  much  increase  of  strength  and  hardness  as 
10  per  cent  of  copper.  Cubical  expansion  by  heat  0.0083,  from  32°  to 
212°  F.  Melts  at  1200°  F.  Specific  heat  0.25. 

Manganese,  Mn.  —  At.  wt.  55.  Sp.  gr.  7  to  8.  The  pure  metal  is  not 
used  in  the  arts,  but  alloys  of  manganese  and  iron,  called  spiegeleisen 
when  containing  below  25  per  cent  of  manganese,  and  ferro-manganese 
when  containing  from  25  to  90  per  cent,  are  used  in  the  manufacture  of 
steel.  Metallic  manganese,  when  alloyed  with  iron,  oxidizes  rapidly  in 
the  air,  and  its  function  in  steel  manufacture  is  to  remove  the  oxygen 
from  the  bath  of  steel  whether  it  exists  as  oxide  of  iron  or  as  occluded 
gas. 

Mercury  (Hydrargyrum),  Hg.  —  At.  wt.  199.8.  A  silver-white  metal, 
liquid  at  temperatures  above  —  39°  F.,  and  boils  at  680°  F.  Unchange- 
able as  gold,  silver,  and  platinum  in  the  atmosphere  at  ordinary  tem- 
peratures, but  oxidizes  to  the  red  oxide  when  near  its  boiling-point. 
Sp.  gr.:  when  liquid  13.58  to  13.59,  when  frozen  14.4  to  14.5.  Easily 
tarnished  by  sulphur  fumes,  also  by  dust,  from  which  it  may  be  freed 
by  straining  through  a  cloth.  No  metal  except  iron  or  platinum  should 
be  allowed  to  touch  mercury.  The  smallest  portions  of  tin,  lead,  zinc, 
and  even  copper  to  a  less  extent,  cause  it  to  tarnish  and  lose  its  perfect 
liquidity.  Coefficient  of  cubical  expansion  from  32°  to  212°  F.  0.0182; 
per  deg.  0.000101. 

Nickel,  Ni.  —  At.  wt.  58.7.  Sp.  gr.  8.27  to  8.93.  A  silvery-white 
metal  with  a  strong  lustre,  not  tarnishing  on  exposure  to  the  air.  Duc- 
tile, hard,  and  as  tenacious  as  iron.  It  is  attracted  to  the  magnet  and 
may  be  made  magnetic  like  iron.  Nickel  is  very  difficult  of  fusion,  melt- 
ing at  about  3000°  F.  Chiefly  used  in  alloys  with  copper,  as  german- 
silver,  nickel-silver,  etc.,  and  also  in  the  manufacture  of  steel  to  increase 
its  hardness  and  strength,  also  for  nickel-plating.  Cubical  expansion 
from  32°  to  212°  F.,  0.0038.  Specific  heat  0.109. 

Platinum,  Pt.  —  At.  wt.  194X  A  whitish  steel-gray  metal,  malleable, 
very  ductile,  and  as  unalterable  by  ordinary  agencies  as  gold.  When 
fused  and  refined  it  is  as  soft  as  copper.  Sp.  gr.  21.15.  It  is  fusible  only 
by  the  oxyhydrogen  blowpipe  or  in  strong  electric  currents.  When  com- 
bined with  iridium  it  forms  an  alloy  of  great  hardness,  which  has  been 
used  for  gun- vents  and  for  standard  weights  and  measures.  The  most 
important  uses  of  platinum  in  the  arts  are  for  vessels  for  chemical  labo- 
ratories and  manufactories,  and  for  the  connecting  wires  in  incandescent 
electric  lamps  and  for  electrical  contact  points.  Cubical  expansion  from 
32°  to  212°  F.,  0.0027,  less  than  that  of  any  other  metal  except  the  rare 
metals,  and  almost  the  same  as  glass. 

Silver  (Argentum),  Ag.  —  At.  wt.  107.9.  Sp.  gr.  10.1  to  11.1,  accord- 
ing to  condition  and  purity.  It  is  the  whitest  of  the  metals,  very  malle- 
able and  ductile,  and  in  hardness  intermediate  between  gold  and  copper. 
Melts  at  about  1750°  F.  Specific  heat  0.056.  Cubical  expansion  from 
32°  to  212°  F.,  0.0058.  As  a  conductor  of  electricity  it  is  equal  to  copper. 
As  a  conductor  of  heat  it  is  superior  to  all  other  metals. 

Tin  (Stannum),  Sn.  —  At.  wt.  119.  Sp.  gr.  7.293.  White,  lustrous, 
soft,  malleable,  of  little  strength,  tenacity  about  3500  Ibs.  per  square 
inch.  Fuses  at  442°  F.  Not  sensibly  volatile  when  melted  at  ordinary 
heats.  Heat  conductivity  14.5,  electric  conductivity  12.4;  silver  being 
100  in  each  case.  Expansion  of  volume  by  heat  0.0069  from  32°  to  212°  F. 
Specific  heat  0.055.  Its  chief  uses  are  for  coating  of  sheet-iron  (called 
tin  plate)  and  for  making  alloys  with  copper  and  other  metals. 

Zinc,  Zn.— At.  wt.  65.4.  Sp.  gr.  7.14.  Melts  at  780°  F.  Volatilizes 
and  burns  in  the  air  when  melted,  with  bluish-white  fumes  of  zinc  oxide. 
It  is  ductile  and  malleable,  but  to  a  much  less  extent  than  copper,  and 


180 


MATERIALS. 


its  tenacity,  about  5000  to  6000  Ibs.  per  square  inch,  is  about  one  tenth 
that  of  wrought  iron.  It  is  practically  non-corrosive  in  the  atmosphere, 
a  thin  film  of  carbonate  of  zinc  forming  upon  it.  Cubical  expansion 
between  32°  and  212°  F.,  0.0088.  Specific  heat  0.096.  Electric  conduc- 
tivity 29,  heat  conductivity  36,  silver  being  100.  Its  principal  uses  are 
for  coating  iron  surfaces,  called  "galvanizing,"  and  for  making  brass  and 
other  alloys. 

Table  Showing  the  Order  of 

Tenacity.  Infusibility. 

Iron  Platinum 

Copper  Iron 

Aluminum  Copper 

Platinum  Gold 

Silver  Silver 

Zinc  Aluminum 

Gold  Zinc 

Tin  Lead 

Lead  Tin 

MEASURES   AND  WEIGHTS   OF    VARIOUS  MATERIALS 
(APPROXIMATE). 


Malleability. 

Ductility. 

Gold 

Platinum 

Silver 

Silver 

Aluminum 

Iron 

Copper 
Tin 

Copper 
Gold 

Lead 

Aluminum 

Zinc 

Zinc 

Platinum 

Tin 

Iron 

Lead 

Brickwork.  —  Brickwork    is    estimated 
various  thicknesses  of  wall  runs  as  follows: 


by    the    thousand,    and    for 


8i/4-in.  wall,  or  1  brick  in  thickness,  14  bricks  per  superficial  foot. 
123/4  "  ••  11/2"  21       " 

17  "         "          "    9          "         "  "  Oft          "  "  "  «• 


17 

2U/2 


28 
35 


An  ordinary  brick  measures  about  81/4X4  X  2  inches,  which  is  equal 
to  66  cubic  inches,  or  26.2  bricks  to  a  cubic  foot.  The  average  weight  is 
4 1/2  Ibs. 

Fuel.  —  A  bushel  of  bituminous  coal  weighs  76  pounds  and  contains 
2688  cubic  inches  =  1.554  cubic  feet.  29.47  bushels  =  1  gross  ton. 

One  acre  of  bituminous  coal  contains  1600  tons  of  2240  pounds  per 
foot  of  thickness  of  coal  worked.  15  to  25  per  cent  must  be  deducted  for 
waste  in  mining. 

41  to  45  cubic  feet  bituminous  coal  when  broken  down  =  1  ton,  2240  Ibs. 


34  t< 
123 
70.9 
1  cu 
1 
1 

a 

i 
i 

3  41 

bic  fo 

'    anthracite  prepared  for  market  .  . 
'    of  charcoal 

.    =  1  ton,  2240  Ibs. 
=  1  ton   2240  Ibs 

"        "    "  coke           .        

.    =  1  ton,  2240  Ibs 

ot  of  anthracite  coal 

=  55  to  66  Ibs 

"  bituminous  coal                                 .  .  . 

«=  50  to  55  Ibs 

Cumberland  (semi-bituminous)  coal.  .  .  . 
Cannel  coal                               .          

=  53  Ibs. 
=  50  3  Ibs 

Charcoal  (hardwood)  

=  18.5  Ibs. 

"         (nine)  .  . 

=  18  Ibs. 

A  bushel  of  coke  weighs  40  pounds  (35  to  42  pounds). 

A  bushel  of  charcoal.  - —  In  1881  the  American  Charcoal-Iron  Work- 
ers' Association  adopted  for  use  in  its  official  publications  for  the  stand- 
ard bushel  of  charcoal  2748  cubic  inches,  or  20  pounds.  A  ton  of  char- 
coal is  to  be  taken  at  2000  pounds.  This  figure  of  20  pounds  to  the 
bushel  was  taken  as  a  fair  average  of  different  bushels  used  throughout 
the  country,  and  it  has  since  been  established  by  law  in  some  States. 

Cement. — Portland,  per  bbl.  net,  376  Ibs.,  per  bag,  net 94  Ibs. 

Natural,  per  bbl.  net,  282  Ibs.,  per  bag  net 94  Ibs. 

Lime. — A  struck  bushel 72  to  75  Ibs. 

Grain. — A  struck  bushel  of  wheat  =  60  Ibs.;  of  corn  =  56  Ibs.;  of 
oats  =  30  Ibs. 

Salt. — A  struck  bushel  of  salt,  coarse,  Syracuse,  N.  Y.  =  56  Ibs.; 
Turk's  Island  =  76  to  80  Ibs. 


MEASUKES  AND   WEIGHTS   OF   VARIOUS   MATERIALS.    181 


Ores,  Earths,  etc. 

13  cubic  feet  of  ordinary  gold  or  silver  ore,  in  mine    =  1  ton  =  2000  Ibs. 

20  "  broken  quartz =1  ton  =  2000  Ibs. 

18  feet  of  gravel  in  bank =1  ton. 

27  cubic  feet  of  gravel  when  dry =1  ton. 

25  "  sand 

18  "  earth  in  bank 

27  "  earth  when  dry 

17  "  clay 

Except  where  otherwise  stated,  a  ton  =  2240  Ibs. 

WEIGHTS   OF  LOGS,   LUMBER,    ETC. 

Weight  of  Green  Logs  to  Scale  1000  Feet,  Board  Measure. 

Yellow  pine  (Southern) 8,000  to  10,0001bs. 


1  ton. 

1  ton. 
=  1  ton. 
=  1  ton. 


, 

Norway  pine  (Michigan) 7,000  to    8,000 

WhitP  ninp  nVTirhiffaTi^  I  off  of  stumP 7'000  to     7,000 

(Micnigan)  j  QUt  of  water 7  000  to    g  000 

White  pine  (Pennsylvania),  bark  off 5,000  to    6,000 

Hemlock  (Pennsylvania),  bark  off . 6,000  to    7,000 

Four  acres  of  water  are  required  to  store  1,000,000  feet  of  logs. 

Weight  of  1000  Feet  of  Lumber,  Board  Measure. 

Yellow  or  Norway  pine Dry,  3,000  Ibs.       Green,  5,000  Ibs. 

White  pine '     2,500    "  4,000    " 

Weight  of  1  Cord  of  Seasoned  Wood,  128  Cu.  Ft.  per  Cord,  Ibs. 


Hickory  or  sugar  maple. .  .  .  4,500 

White  oak 3,850 

Beech,  red  oak  or  black  oak  .  3,250 


Poplar,  chestnut  or  elm. .  .  2,350 
Pine  (white  or  Norway)..  .  2,000 
Hemlock  bark,  dry 2,200 


WEIGHT  OF  RODS,   BARS,   PLATES,   TUBES,   AND   SPHERES 
OF   DIFFERENT   MATERIALS. 

Notation:  b  =  breadth,  t  =  thickness,  s  =  side  of  square,  D  =  ex- 
ternal diameter,  d  =  internal  diameter,  all  in  inches. 

Sectional  areas:  of  square  bars  =  s2;  of  flat  bars  =  W;  of  round  rods 
=  0.7854  Z>2;  of  tubes  •=  0.7854  (D2  -  rf2)  =  3.1416  (Dt  -Z2). 

Volume  of  1  foot  in  length:  of  square  bars  =  12s2;  of  flat  bars  =  12bt; 
of  round  bars  =  9.4248D2;  of  tubes  =  9.4248  (D2  -  d2)  =  37.699 
(Dt  -22),  in  cu.  in. 

Weight  per  foot  length  =  volume  +  weight  per  cubic  inch  of  mate- 
rial. Weight  of  a  sphere  =  diam.3  X  0.5236  X  weight  per  cubic  inch. 


3 

$& 

r.g 

. 

4 

. 

d 

u  . 

*?  rX 

[Vj    f^.O 

fe  PQ 

U    . 

j>  -fj  iH 

JV   'tf      » 

Material. 

P| 

5  > 

^ 

fctrf 

g| 

If  fa 

y  • 

"gjj 

JJ 

ge 

g-w 

+i«*H   ^ 

i*s 

i* 

I^-S 

£-3« 

4J  a 

82  X 

btX 

D*X 

D*X 

Cast  iron  

7.218 

450. 

37.5 

31/8 

31/8 

0.2604 

15-16 

2.454 

0.1363 

Wrought  iron.  . 

7.7 

480. 

40. 

31/3 

31/3 

.2779 

1. 

2.618 

.1455 

Steel 

7.854 

489.6 

40.8 

3.4 

3.4 

.2833 

1.02 

2.670 

.1484 

Copper  &  Bronze 
(copper  and  tin) 

8.855 

552. 

46. 

3.833 

3.833 

.3195 

1.15 

3.011 

.1673 

Brass  (  £  zm^* 

8.393 

523.2 

43.6 

3.633 

3.633 

«3029 

1.09 

2.854 

.1586 

Monel  metal,  rolled 

8.95 

558. 

46.5 

3.87 

3.87 

.323 

1.16 

3.043 

.1691 

Lead... 

1  1.38 

709.6 

59.1 

493 

493 

.4106 

1  48 

3.870 

.2150 

Aluminum  

2.67 

166.5 

13.9 

1.16 

1.16 

.0963 

0.347 

0.908 

.0504 

Glass. 

2.62 

163.4 

13.6 

1.13 

1  13 

.0945 

0.34 

0.891 

.0495 

Pine  wood,  dry  

0.481 

30.0 

2.5 

0.21 

0.21 

.0174 

1-16 

0.164 

.0091 

Weight  per  cylindrical  in.,  1  in.  long,  =  coefficient  of  D2  in  next  to 
last  column  -7- 12. 


182  MATERIALS. 

FOP  tubes  use  the  coefficient  of  D2  in  next  to  last  column,  as  for  rods, 
and  multiply  it  into  (D2  —  d2) ;  or  multiply  it  by  4  (Dt  -  22) . 

For  hollow  spheres  use  the  coefficient  of  D3  in  the  last  column  and 
multiply  it  into  (D3  -  d3). 

For  hexagons  multiply  the  weight  of  square  bars  by  0.866  (short 
diam.  of  hexagon  =  side  of  square).  For  octagons  multiply  by  0.8284. 

COMMERCIAL    SIZES    OF  MERCHANT  IRON  AND  STEEL 

BARS. 

Steel  Bars. 

Flats,  Square  Edge. — s/g  to  3  in.  wide,  by  any  thickness  from 
1/8  in.  up  to  width;  3  to  5  in.  wide  by  any  thickness  1/4  t9  3  in. 
inclusive;  5  to  7  in.  wide,  by  any  thickness,  1/4  to  2  in.  inclusive. 

Flats,  Band  Edge. — Thicknesses  are  in  B.  W.  G.,  3/8  in.  wide  by 
No.  18  to  No.  4.  7/i6  in.  by  No.  19  to  No.  4.  1/2  in.  by  No.  22  to  No. 
4.  9/i6  to  1  in.  by  No.  23  to  No.  4.  1 1/16  to  2  in.  by  No.  22  to  No.  4. 
2Vi6  to  3  in.  by  No.  21  to  No.  1.  39/16  to  4  in.  by  No.  19  to  No.  1. 
4Vi6  to  41/2  in.  by  No.  18  to  No.  1.  49/i6  to  5  Vie  in.  by  No.  17  to  No.  1. 
5 i/s  to  6 3/4  in.  by  No.  16  to  No.  1.  7  in.,  7 1/4  in.,  7 1/2  in.,  7  5/8  in.,  7 3/4  in., 
7 7/8  in.,  8  in.,  81/4  in.,  81/2  in.,  85/8  in.,  each  by  No.  14  to  No.  1.  95/8 
in.  by  No.  12  to  No.  1. 

Squares. — Widths  across  faces:  3/ie  to  2  in.,  advancing  by  1/64  in.; 
21/32  to  3  1/2  in.,  advancing  by  1/32  in.;  3  9/ie  to  51/2  in.,  advancing  by 
Vie  in. 

Round-cornered  Squares. — 1/4  to  3/4  in.,  across  faces,  advancing 
by  1/64  in. 

Rounds. — Diameters:  7/32  to  13/4  in.,  inclusive,  advancing  by  1/64 
in.;  1  25/32  in.  to  31/2  in.  inclusive,  advancing  by  1/32;  3  9/ie  to  7  in., 
inclusive,  advancing  by  Vie  in. 

Half  Rounds. — Diameters:  5/16  to  7/s  in.,  inclusive,  advancing  by 
1/64  in. ;  15/16  to  1 3/4  in . ,  advancing  by  Vie  in. ;  2  in. ;  2 1/2  in. ;  3  in. 

.Hexagons. — Width  across  faces:  1/4  to  13/ie  in.,  inclusive,  advanc- 
ing by  1/32  in.;  1  1/4  in.  to  3 Vie  in.,  advancing  by  Vie  in. 

Iron  Bars. 

Round. — 3/i6  to  1 7/8  in.,  advancing  by  Vs2  in.;  1 15/i6  to  2 3/4  in.,  advancing 
by  Vie  in.;  2  7/8  to  3  3/4  in.,  advancing  by  Vs  in.;  4  to  5  in.,  advancing  by 
1/4  in. 

Squares. — Vie  to  5/s  in.,  advancing  by  1/32  in.;  n/ie  in.  to  1  in.,  advancing 
by  Vie  in.;   1  Vg  in.  to  2  1/2  in.,  advancing  by  Vs  in.;  2  3/4  in.  to  4  */2  in.,  ad- 
vancing by  1/4  in- 
Half  Rounds.— -8/g,   7/16,   l/2,   5/8,   11/16,   3/4,   7/8f   1,   1  l/g,    1  \j£\  3/g,    1  l/2, 

1 3/4,  2  in. 

OvalS.— V2  X  V4,   5/8  X  5/16,  3/4  X  3/8  and  7/8  X  7/16  in. 


Half  Ovals.— 1/2  X  Vie,  Vs  X  Vie,  3/4  X  Vie,  Vs  X  Vie,  1  X  Vie, 
3/4  X  V4,  Vs  X  i/4,  1  X  i/4,  1  Vs  X  1/4.  1  X  Vie,  1  Vs  X  Vie,  1  V4  X  Vie, 
1  X  Vs,  1  Vs  X  Vs,  1  V4  X  Vs,  1  V2  X  Vs,  1  3/4  X  V*  2  X  Vs  in. 


Flats.— 1/2  X  Vie  to  Vs  in.;  Vs  X  Vie  to  1/2  in.;  3/4  X  Vie  to  Vs  in.; 
Vs  X  Vie  to  3/4  in.;  1  X  Vie  to  Vs  in.;  1  Vie  X  i/4  to  Vs  in-;  1  Vs  X  Vie  to 
1  in.;  1  1/4  X  Vie  to  1  in.;  1  3/s  X  Vie  to  1  Vs  in.;  1  1/2  X  Vie  to  1  1/4  in.; 

1  Vs  X  V4  to  1  1/2  in.;   1  3/4  X  Vie  to  1  1/2  in.;   1  Vs  X  I/A  to  1  1/2  in.;  2  X  Vie 
to  1  3/4  in.;  2  Vs  X  V4  to  1  1/4  in.;  2  i/4  X  Vie  to  2  in.;  2  Vs  X  V4  to  1  3/4  in.; 
2V2  XVie  to  2V4  in.;    2  Vs  X  V4  to  2  i/4  in.;    2  3/4  X  Vie  to  2  1/2  in.; 

2  7/8  X  V8  to  1/2  in.;  2  Vs  X  Vs  to  2  i/4  in.;  3  X  Vie  to  2  3/4  in.;  3  Vs  X  1  V2 
to  2  Vs  in.;  3 1/4  X  */4  to  2 3/4  in.;  3  1/2  X  Vie  to  2  Vs  in.;  3  3/4  X  V4  to  3  in.; 

•4  X  V4  to  3  in.;  4  1/4  X  V4  to  2  in.;  4  1/2  X  J/4  to  2  1/2  in.;  4  3/4  X  V4  to  2 
in.;  5  X  x/4  to  2  3/4  in.;  5  1/2  X  V4  to  2  in.;  6  X  V4  to  2  in.;  6  1/2  X  V4  to 
1  in.;  7  X  !/4  to  2  in.;  7  1/2  X  x/4  to  1  in.;  8  X  V4  to  2  in. 

Round  Edge  Flats. — 1  to  2  in.  wide  by  V4  to  1  V4  in.  thick;  2 1/4  to 
4  1/2  in.  wide  by  3/s  to  1 1/4  in.  thick. 


WEIGHT   OP  IRON   AND   STEEL   SHEETS. 


183 


WEIGHT  OF  IRON  AND  STEEI,  SHEETS. 

Weights  in  Pounds  per  Square  Foot. 

(For  weights  by  the  Decimal  Gauge,  see  page  33.) 


Thickness  by  Birmingham  Gauge. 

U.  S.  Standard  Gauge,  1893.     (See 
p.  32.) 

No.  of 
Gauge. 

Thick- 
ness in 
Inches. 

Iron. 

Steel. 

No.  of 
Gauge. 

Thick- 
ness, In. 
(Approx.) 

Iron. 

Steel. 

0000 

0.454 

18.16 

18.52 

0000000 

0.5 

20. 

20.40 

000 

.425 

17.00 

17.34 

000000 

0.4688 

18.75 

19.125 

00 

.38 

15.20 

15.50 

00000 

0.4375 

17.50 

17.85 

0 

.34 

13.60 

13.87 

0000 

0.4063 

16.25 

16.575 

1 

.3 

12.00 

12.24 

000 

0.375 

15. 

15.  3C 

2 

.284 

11.36 

11.59 

00 

0.3438 

13.75 

14.025 

3 

.259 

10.36 

10.57 

0 

0.3125 

12.50 

12.75 

4 

.238 

9.52 

9.71 

1 

0.2813 

11.25 

11.475 

5 

.22 

8.80 

8.98 

2 

0.2656 

10.625 

10.837 

6 

.203 

8.12 

8.28 

3 

0.25 

10. 

10.20 

7 

.18    ' 

7.20 

7.34 

4 

0.2344 

9.375 

9.562 

8 

.165 

6.60 

6.73 

5 

0.2188 

8.75 

8.925 

9 

.148 

5.92 

6.04 

6 

0.2031 

8.125 

8.287 

10 

.134 

5.36 

5.47 

7 

0.1875 

7.5 

7.65 

11 

.12 

4.80 

4.90 

8 

0.1719 

6.875 

7.012 

12 

.109 

4.36 

4.45 

9 

0.1563 

6.25 

6.375 

13 

.095 

3.80 

3.88 

10 

0.1405 

5.625 

5.737 

14 

.083 

3.32 

3.39 

11 

0.125 

5. 

5.10 

15 

.072 

2.88 

2.94 

12 

0.1094 

4.375 

4.462 

16 

.065 

2.60 

2.65 

13 

0.0938 

3.75 

3.825 

17 

.058 

2.32 

2.37 

14 

0.0781 

3.125 

3.187 

18 

.049 

.96 

2.00 

15 

0.0703 

2.8125 

2.869 

19 

.042 

.68 

1.71 

16 

0.0625 

2.5 

2.55 

20 

.035 

.40 

1.43 

17 

0.0563 

2.25 

2.295 

21 

.032 

.28 

1.31 

18 

0.05 

2. 

2.04 

22 

.028 

.12 

1.14 

19 

0.0438 

.75 

.785 

23 

.025 

.00 

1.02 

20 

0.0375 

.50 

.53 

24 

.022 

.88 

.898 

21 

0.0344 

.375 

.402 

25 

.02 

.80 

.816 

22 

0.0312 

.25 

.275 

26 

.018 

.72 

.734 

23 

0.0281 

.125 

.147 

27 

.016 

.64 

.653 

24 

0.025 

.02 

28 

.014 

.56 

.571 

25 

0.0219 

0^875 

0.892 

29 

.013 

.52 

.530 

26 

0  0188 

0.75 

0.765 

30 

.012 

.48 

.490 

27 

0.0172 

0.6875 

0.701 

31 

.01 

.40 

.408 

28 

0.0156 

0.625 

0.637 

32 

.009 

.36 

.367 

29 

0.0141 

0.5625 

0.574 

33 

.008 

.32 

.326 

30 

0.0125 

0.5 

0.51 

34 

.007 

.28 

.286 

31 

0.0109 

0.4375 

0.446 

35 

.005 

.20 

.204 

32 

0.0102 

0.40625 

0.414 

36 

.004 

.16 

.163 

33 

0.0094 

0.375 

0.382 

34 

0.0086 

0.34375 

0.351 

35 

0.0078 

0.3125 

0.319 

36 

0.0070 

0.28125 

0.287 

37 

0.0066 

0.26562 

0.271 

38 

0.0063 

0.25 

0.255 

Iron.           Steel. 

Specific  gravity  .  .                                7.7                7.854 

489.6 
Weight  per  cubic  inch 0.2778          0.2833 

As  there  are  many  gauges  in  use  differing  from  each  other,  and  even  the 
thicknesses  of  a  certain  specified  gauge,  as  the  Birmingham,  are  not  assumed 
the  same  by  all  manufacturers,  orders  for  sheets  and  wires  should  always 
state  the  weight  per  square  foot,  or  the  thickness  in  thousandths  of  an  inch. 


184 


MATERIALS. 


WEIGHTS  OF  SQUARE  AND  ROUND  BARS  OP  WROUGHT 
IRON  IN  POUNDS   PER  LINEAL  FOOT. 

Iron  weighing  480  Ib.  per  cubic  foot.     For  steel  add  2  per  cent. 


Thickness  or 
Diameter 
in  Inches. 

2l? 

Sgj 
*& 

°ii 

!§e^ 

£J! 

Thickness  or 
Diameter 
in  Inches. 

*H  e8  M 
5Ǥ 

•ajM 

'8  3^ 

*Jt 

Weight  of 
Round  Bar 
1  Ft.  Long. 

Thickness  or 
Diameter 
in  Inches. 

!lf 
»IJ 
II! 

Weight  of 
Round  Bar  I 
1  Ft.  Long.  [ 

0 

H/16 

24.08 

18.91 

3/8 

96.30 

75.64 

Vl6 

0.013 

0.010 

3/4 

25.21 

19.80 

7/16 

08.55 

77.40 

VS 

.052 

.041 

13/16 

26.37 

20.71 

1/2 

100.8 

79.19 

3/16 

.117 

.092 

7/8 

27.55 

21.64 

/16 

103.1 

81.00 

1/4 

.208 

.164 

15/16 

28.76 

22.59 

5/8 

105.5 

82.83 

5/16 

.326 

.256 

3 

30.00 

23.56 

H/16 

107.8 

84  69 

3/8 

.469 

.368 

1/16 

31.26 

24.55 

3/4 

110.2 

86.56 

7/16 

.638 

.501 

1/8 

32.55 

25.57 

13/16 

112.6 

88.45 

>/2 

.833 

.654 

3/16 

33.87 

26.60 

7/8 

115.1 

9036 

9/16 

1.055 

.828 

1/4 

35.21 

27.65 

15/16 

117.5 

92.29 

5/8 

1.302 

1.023 

5/16 

36.58 

28.73 

6 

120.0 

94.25 

H/16 

1.576 

1.237 

3/8 

37.97 

29.82 

1/8 

125.1 

98.22 

3/4 

1.875 

1.473 

7/16 

39.39 

30.94 

1/4 

130.2 

102.3 

13/16 

2.201 

1.728 

1/2 

40.83 

32.07 

3/8 

135.5 

106.4 

7/8 

2.552 

2.004 

9/16 

42.30 

33.23 

V2 

140.8 

110.6 

15/16 

2.930 

2.301 

5/8 

43.80 

34.40 

5/8 

146.3 

114.9 

1 

3.333 

2.618 

H/16 

45.33 

35.60 

3/4 

151.9 

119.3 

Vl6 

3.763 

2.955 

3/4 

46.88 

36.82 

7/8 

157*6 

123.7 

1/8 

4.219 

3.313 

13/16 

48.45 

38.05 

163.3 

128.3 

3/16 

4.701 

3.692 

7/8 

50.05 

39.31 

1/8 

169.2 

132.9 

1/4 

5.208 

4.091 

15/16 

51.68 

40.59 

1/4 

175.2 

137.6 

5/16 

5.742 

4.510 

4 

53.33 

41.89 

3/8 

181.3 

1424 

3/8 

6.302 

4.950 

1/16 

55.01 

43.21 

1/2 

187  5 

147.3 

7/16 

6.888 

5.410 

1/8 

56.72 

44.55 

5/8 

193.8 

152.2 

1/2 

7.500 

5.890 

3/16 

58.45 

45.91 

3/4 

200.2 

157.2 

9/16 

8.138 

6.392 

1/4 

60.21 

47.29 

7/8 

206.7 

162.4 

5/8 

8.802 

6.913 

5/16 

61.99 

48.69 

213.3 

167.6 

H/16 

9.492 

7.455 

3/8 

63.80 

50.11 

1/4 

226.9 

178.2 

3/4 

10.21 

8.018 

7/16 

65.64 

51.55 

1/2 

240.8 

189.2 

13/16 

10.95 

8.601 

V2 

67.50 

53.01 

3/4 

255.2 

200.4 

7/8 

11.72 

9.204 

9/16 

69.39 

54.50 

9 

270.0 

212.1 

15/16 

12.51 

9.828 

5/8 

.71.30 

56.00 

1/4 

285.2 

224.0 

2 

13.33 

10.47 

U/16 

73.24 

57.52 

1/9 

300.8 

236.3 

1/16 

14.18 

11.14 

3/4 

75.21 

59.07 

3/4 

316.9 

248.9 

1/8 

15.05 

11.82 

13/16 

77.20 

60.63 

10 

333.3 

261.8 

3/16 

15.95 

12.53 

7/8 

79.22 

62.22 

1/4 

350.2 

275.1 

1/4 

16.88 

13.25 

15/16 

81.26 

63.82 

1/9 

367.5 

288.6 

5/16 

17.83 

14.00 

5 

83.33 

65.45 

3/4 

385.2 

302.5 

3/8 

18.80 

14.77 

Vl6 

85.43 

67.10 

11 

403  3 

3168 

7/16 

1980 

15.55 

1/8 

87.55 

68.76 

1/4 

421.9 

331.3 

1/2 

20.83 

16.36 

3/16 

89.70 

70.45 

1/2 

440.8 

346.2 

9/16 

21.89 

17.19 

1/4 

91.88 

72.16 

3/4 

460.2 

361.4 

5/8 

22.97 

18.04 

5/16 

94.08 

73.89 

12 

480. 

377. 

WEIGHT  OF  STEEL   BARS. 


185 


WEIGHT  OP  SQUARE  AND  ROUND  STEEL  BARS   PER   LINEAL 
FOOT.      (Steel  Weighing  489.6  Ib.  per  cu.  ft.) 


Thickness  or 
Diameter 
in  Inches. 

Weight  of 
Square  Bar 
1  Ft.  Long. 

Weight  of 
Round  Bar 
1  Ft.  Long. 

Thickness  or 
Diameter 
in  Inches. 

Weight  of 
Square  Bar 
1  Ft.  Long. 

°«*e 
•s    s 
Sl% 

!§£ 
^«- 

Thickness  or 
Diameter 
in  Inches. 

Weight  of 
Square  Bar 
1  Ft.  Long. 

Weight  of 
Round  Bar 
1  Ft.  Long. 

0 

H/16 

24.56 

19.29 

3/8 

98.23 

77.15 

1/16 

0.013 

0.010 

3/4 

25.71 

20.20 

7/16 

100.5 

78.95 

1/8 

.053 

.042 

13/16 

26.90 

21.12 

1/2 

102.8 

80.77 

3/16 

.119 

.094 

7/8 

28.10 

22.07 

9/16 

105.2 

82.62 

V4 

.212 

.167 

15/16 

29.34 

23.03 

5/8 

107.6 

84.49 

5/16 

.333 

.261 

3 

30.60 

24.03 

U/ifl 

110.0 

86.38 

3/8 

.478 

.375 

1/16 

31  .89 

25.04 

3/4 

112.4 

88.29 

7/16 

.651 

.511 

1/8 

33.20 

26.08 

13/16 

114.9 

90.22 

1/2 

.850 

.667 

3/16 

34.55 

27.13 

7/8 

117.4 

92.17 

9/16 

1.076 

.845 

1/4 

35.91 

28.20 

15/16 

119.9 

94.14 

5/8 

1.328 

1  .043 

5/16 

37.31 

29.30 

6 

122.4 

96.14 

H/16 

1.608 

1.262 

3/8 

38.73 

30.42 

1/8 

127.6 

100.2 

3/4 

1  .913 

1  .502 

7/16 

40.18 

31  .56 

1/4 

132.8 

104.3 

13/16 

2.245 

1.763 

1/2 

41  .65 

32.71 

3/8 

138.2 

108.5 

7/8 

2.603 

2.044 

9/16 

43.15 

33.89 

1/2 

143.6 

112.8 

15/16 

2.989 

2.347 

5/8 

44.68 

35.09 

5/8 

149.2 

117.2 

1 

3.400 

2.670 

U/16 

46.24 

36.31 

3/4 

154.9 

121.7 

1/16 

3.838 

3.014 

3/4 

47.82 

37.56 

7/8 

160.8 

126.2 

1/8 

4.303 

3.379 

13/16 

49.42 

38.81' 

7 

166.6 

130.9 

3/16 

4.795 

3.766 

7/8 

51  .05 

40.10 

1/8 

172.6 

135.6 

1/4 

5.312 

4.173 

15/16 

52.71 

41  .40 

1/4 

178.7 

140.  < 

5/16 

5.857 

4.600 

4 

54.40 

42.73 

3/8 

184.9 

145.1 

3/8 

6.428 

5.049 

1/16 

56.11 

44.07 

1/2 

191.3 

150.2 

7/16 

7.026 

5.518 

1/8 

57.85 

45.44 

5/8 

197.7 

155.2 

1/2 

7.650 

6.008 

3/16 

59.62 

46.83 

3/4 

204.2 

159.3 

9/16 

8.301 

6.520 

1/4 

61:41 

48.24 

7/8 

210.8 

165.6 

5/8 

8.978 

7.051 

5/16 

63.23 

49.66 

8 

217.6 

171.0 

H/16 

9.682 

7.604 

3/8 

65.08 

51.11 

1/4 

231.4 

181.8 

3/4 

10.41 

8.178 

7/16 

66.95 

52.58 

1/2 

245.6 

193.0 

13/16 

11  .17 

8.773 

1/2 

68.85 

54.07 

3/4 

260.3 

204  .4 

7/8 

11  .95 

9.388 

9/16 

70.78 

55.59 

9 

275.4 

216.3 

15/16 

12.76 

10.02 

5/8 

72.73 

57.12 

1/4 

290.9 

228.5 

2 

13.60 

10.68 

n/i6 

74.70 

58.67 

1/2 

306.8 

241.0 

1/16 

14.46 

11  .36 

3/4 

76.71 

60.25 

3/4 

323.2 

253.9 

1/8 

15.35 

12.06 

13/16 

78.74 

61.84 

10 

340.0 

267.0 

3/16 

16.27 

12.78 

7/8 

80.80 

63.46 

1/4 

357.2 

280.6 

1/4 

17.22 

13.52 

15/16 

82.89 

65.10 

1/2 

374.9 

294.  4 

5/16 

18.19 

14.28 

5 

85.00 

66.76 

3/4 

392.9 

308.6 

3/8 

19.18 

15.07 

Vl6 

87.14 

68.44 

11 

411.4 

323.1 

7/16 

20.20 

15.86 

'   1/8 

89.30 

70.14 

1/4 

430.3 

337.9 

1/2 

21.25 

16.69 

3/16 

91  .49 

71.86 

1/2 

449.6 

353.1 

9/16 

22.33 

17.53 

1/4 

93  72 

73.60 

3/4 

469.4 

368.6 

5/8 

23.43 

18.40 

5/16 

95.96 

75.37 

12 

489.6 

384.5 

Weight  of  Fillets. 


Ra- 
dius, 
In. 

Area, 
Sq.  In. 

Weight  per  In.,  Lb. 

Ra- 
dius, 
In. 

Area, 
Sq.  In. 

Weight  per  In.,  Lb. 

Cast 
Iron. 

Steel. 

Brass. 

Cast 
Iron. 

Steel. 

Brass. 

1/4 

0.0134 

0.0035 

0.0038 

0.0040 

13/16 

0.1416 

0.0369 

0.0401 

0.0414 

5/16 

.0209 

.0054 

.0059 

.0061 

7/8 

.1634 

.0428 

.0465 

.0479 

3/8 

.0302 

.0078 

.0085 

.0088 

15/16 

.1886 

.0491 

.0534 

.0550 

7/16 

.0411 

.0107 

.0116 

.0120 

1 

.2146 

.0559 

.0608 

.0626 

1/2 

.0536 

.0140 

.0152 

.0157 

1  1/8 

.2716 

.0709 

.0771 

.0794 

9/1  fi 

.0679 

.0177 

.0192 

.0200 

1  1/4 

.3353 

.0874 

.0950 

.0979 

5/8 

.0834 

.0218 

.0237 

.0244 

1  3/8 

.4057 

.0920 

.1000 

.1030 

H/16 

.1014 

.0264 

.0287 

.0300 

1  1/2 

.4828 

.1259 

.1368 

.1410 

3/4 

.1207 

.0315 

.0342 

.0352 

15/8 

.5668 

.1479 

.1608 

.1657 

Continued  on  next  page. 


186 


MATERIALS. 


Weights  per  Lineal  Inch  of  Bound,  Square  and  Hexagon  Steel. 

Weight  of  1  cu.  in.  =  0.2836  Ib.  Weight  of  1  cu.  ft.  »  490  Ib. 


Thick- 
ness or 
Diam- 
eter. 

Round. 

Square. 

Hexagon 

Thick- 
ness or 
Diam- 
eter. 

Round. 

Square. 

Hexagon. 

V32 

0.0002 

0.0003 

0.0002 

17/8 

0  .  783  1 

0.9970 

0.8635 

1/16 

.0009 

.0011 

.0010 

115/ 

.8361 

.0646 

.9220 

3/32 

.0020 

.0025 

.0022 

2 

.8910 

.1342 

.9825 

1/8 

.0035 

.0044 

.0038 

21/16 

.9475 

.2064 

.0448 

5/32 

.0054 

.0069 

.0060 

21/8 

.0058 

.2806 

.1091 

3/16 

.0078 

.0101 

.0086 

23/i6 

.0658 

.3570 

.1753 

7/32 

.0107 

.0136 

.0118 

21/4 

.1276 

.4357 

.2434 

1/4 

.0139 

.0177 

.0154 

25/ie 

.1911 

.5165 

.3135 

9/32 

.0176 

.0224 

.0194 

23/8 

.2564 

.6569 

.3854 

5/16 

.0218 

.0277 

.0240 

27/ie 

.3234 

.6849 

.4593 

H/32 

.0263 

.0335 

.0290 

21/2 

.3921 

.7724 

.5351 

3/8 

.0313 

.0405 

.0345 

25/8 

.5348 

1.9541 

.6924 

13/32 

.0368 

.0466 

.0405 

23/i 

.6845 

2.1446 

.8574 

7/16 

.0426 

.0543 

.0470 

27/8 

.8411 

2.3441 

2.0304 

15/32 

.0489 

.0623 

.0540 

3 

2.0046 

2.5548 

2.2105 

1/2 

.0557 

.0709 

.0614 

31/8 

2.1752 

2.7719 

2.3986 

17/32 

.0629 

.0800 

.0693 

31/4 

2.3527 

2.9954 

2.5918 

9/16 

.0705 

.0897 

.0777 

33/g 

2.5371 

3.2303 

2.7977 

19/32 

.0785 

.1036 

.0866 

31/2 

2.7286 

3.4740 

3.0083 

5/8 

.0870 

.1108 

.0959 

35/8 

2.9269 

3.7265 

3.2275 

21/32 

.0959 

.1221 

.1058 

33/4 

3.1323 

3.9880 

3.4539 

11/16 

.1053 

.1340 

.1161 

37/g 

3.3446 

4.2582 

3.6880 

23/32 

.1151 

.1465 

.1270 

4 

3.5638 

4.5374 

3.9298 

3/4 

.1253 

.1622 

.1382 

41/8 

3.7900 

4.8254 

4.1792 

25/32 

.1359 

.1732 

.1499 

41/4 

4.0232 

5.1223 

4.4364 

13/16 

.1470 

.1872 

.1620 

43/8 

4.2634 

5.4280 

4.7011 

27/32 

.1586 

.2019 

.1749 

41/2 

4.5105 

5.7426 

4.9736 

7/8 

.1705 

.2171 

.1880 

45/8 

4.7645 

6.0662 

5.2538 

29/32 

.1829 

.2329 

.2015 

43/4 

5.0255 

6.6276 

5.5416 

15/16 

.1958 

.2492 

.2159 

47/8 

5.2935 

6.7397 

5.8371 

31/32 

.2090 

.2661 

.2305 

5 

5.5685 

7.0897 

6.1403 

9 

.2227 

.2836 

.2456 

51/8 

5.8504 

7.4496 

6.4511 

1  1/16 

.2515 

.3201 

.2773 

51/4 

6.1392 

7.8164 

6.7697 

1  1/8 

.2819 

.3589 

.3109 

53/8 

6.4351 

8.1930 

7.0959 

1  3/16 

.3141 

.4142 

.  .3464 

51/2 

6.7379 

8.5786 

7.4298 

U/4 

.3480 

.4431 

.3838 

55/8 

7.0476 

8.9729 

7.7713 

1  5/16 

.3837 

.4885 

.4231 

53/4 

7.3643 

9.3762 

8.1214 

1  3/8 

.4211 

.5362 

.4643 

57/8 

7.6880 

9  .  7883 

8.4774 

1  7/16 

.4603 

.5860 

.5076 

6 

8.0186 

10.2192 

8.8420 

1  V2 

.5012 

.6487 

.5526 

61/4 

8  .  7007 

11.0877 

9.5943 

1  9/16 

.5438 

.6930 

.5996 

6l/2 

9.4107 

11.9817 

10.3673 

1  5/8 

.5882 

.7489 

.6480 

63/4 

10.1485 

12.9211 

11.1908 

1  H/16 

.6343 

.8076 

.6994 

7 

10.9142 

13.8960 

12.0351 

1  3/4 

.6821 

.8685 

.7521 

71/2 

12.5291 

15.9520 

13.8158 

1  13/16 

.7317 

.9316 

.8069 

8 

14.2553 

18.1497 

15.7192 

Weight  of  Fillets.— Continued  from  page  185.   . 


Ra- 
dius, 
In. 

Area, 
Sq.  In. 

Weight  per  In.,  Lb. 

Ra- 
dius, 
In. 

Area, 
Sq.  In. 

Weight  per  In.,  Lb. 

Cast 
Iron. 

Steel. 

Brass. 

Cast 
Iron. 

Steel. 

Br.ass. 

13/4 

0.6572 

0.1713 

0.1862 

0.1920 

27/8 

1.774 

0.4621 

0.5022 

0.5017 

1  7/8 

.7545 

.1970 

.2137 

.2202 

3 

1.931 

.4950 

.5471 

.5635 

2 

.8585 

.2237 

.2431 

.2504 

31/4 

2.267 

.5903 

.6417 

.6609 

21/8 

.9692 

.2502 

.2743 

.2826 

31/2 

2.629 

.6926 

.7438 

.7661 

US 

1.086 

.2832 

.3079 

.3172 

33/4 

3.018 

.7873 

.8523 

.8817 

23/8 

1.210 

.3155 

.3429 

.3532 

3.434 

.8933 

.9709 

1.000 

21/2 

1.341 

.3496 

.3800 

.3914 

41/4 

3.876 

1.008 

1.096 

1.130 

25/8 

1.478 

.3857 

.4192 

.4317 

41/2 

4.346 

1.132 

1.231 

1.270 

23/4 

1.623 

.4222 

.4589 

.4727 

43/4 

4.842 

1.261 

1.371 

1.421 

WEIGHT   OF   PLATE   IKON. 


187 


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MATERIALS. 


WEIGHTS  OF  STEEL  BLOOMS. 

Soft  steel.     1  cubic  inch  —  0.284  Ib.     1  cubic  foot  =  490.75  Ibs. 


Size, 
Inches 

Lengths. 

1" 

6" 

13" 

18" 

24" 

30" 

36" 

42" 

48" 

54" 

60" 

66" 

12  X6 
X5 

20.45 
17.04 

123 
102 

245 
204 

368 
307 

491 
409 

613 
511 

736 
613 

~859 
716 

982 

818 

1104 
920 

1227 
1022 

1350 
1125 

X4 

13.63 

82 

164 

245 

327 

409 

491 

573 

654 

736 

818 

900 

11   X6 

18.75 

113 

225 

338 

450 

563 

675 

788 

900 

1013 

1  125 

1238 

X5 

15.62 

94 

188 

281 

375 

469 

562 

656 

750 

843 

937 

1031 

X4 

12.50 

75 

150 

225 

300 

375 

450 

525 

600 

675 

750 

825 

10  X8 

22.72 

136 

273 

409 

545 

682 

818 

954 

1091 

1227 

1363 

1500 

X7 

19.88 

120 

239 

358 

477 

596 

715 

835 

955 

1074 

1193 

1312 

X6 

17.04 

102 

204 

307 

409 

511 

613 

716 

818 

920 

1022 

1125 

X5 

14.20 

85 

170 

256 

341 

426 

511 

596 

682 

767 

852 

937 

X4 

11.36 

68 

136 

205 

273 

341 

409 

477 

546 

614 

682 

750 

K3 

8.52 

51 

102 

153 

204 

255 

306 

358 

409 

460 

511 

562 

9  X8 

20.45 

123 

245 

368 

491 

613 

736 

859 

982 

1104 

1227 

1350 

X7 

17.89 

107 

215 

322 

430 

537 

644 

751 

859 

966 

1073 

1181 

X6 

15.34 

92 

184 

276 

368 

460 

552 

644 

736 

828 

920 

1012 

X5 

12.78 

77 

153 

230 

307 

383 

460 

537 

614 

690 

767 

844 

X4 

10.22 

61 

123 

184 

245 

307 

368 

429 

490 

552 

613 

674 

X3 

7.66 

46 

92 

138 

184 

230 

276 

322 

368 

414 

460 

506 

8  X8 

18.18 

109 

218 

327 

436 

545 

655 

764 

873 

982 

1091 

1200 

X7 

15.9 

95 

191 

286 

382 

477 

572 

668 

763 

859 

954 

1049 

X6 

13.63 

82 

164 

245 

327 

409 

491 

573 

654 

736 

818 

900 

X5 

11.36 

68 

136 

205 

273 

341 

409 

477 

546 

614 

682 

750 

X4 

9.09 

55 

109 

164 

218 

273 

327 

382 

436 

491 

545 

600 

X3 

6.82 

41 

82 

123 

164 

204 

245 

286 

327 

368 

409 

450 

7  X7 

13.92 

83 

167 

251 

334 

418 

501 

585 

668 

752 

835 

919 

X6 

11.93 

72 

143 

215 

286 

358 

430 

501 

573 

644 

716 

788 

X5 

9.94 

60 

119 

179 

7,38 

298 

358 

417 

477 

536 

596 

656 

X4 

7.95 

48 

96 

143 

191 

239 

286 

334 

382 

429 

477 

525 

X3 

5.96 

36 

72 

107 

143 

179 

214 

250 

286 

322 

358 

393 

61/2X61/2 

12. 

72 

144 

216 

288 

360 

432 

504 

576 

648 

720 

792 

X4 

7.38 

44 

89 

133 

177 

221 

266 

310 

354 

399 

443 

487 

6  X6 

10.22 

61 

123 

184 

245 

307 

368 

429 

490 

551 

613 

674 

X5 

8.52 

51' 

102 

153 

204 

255 

307 

358 

409 

460 

511 

562 

X4 

6.82 

41 

82 

123 

164 

204 

245 

286 

327 

368 

409 

450 

X3 

5.11 

31 

61 

92 

123 

153 

184 

214 

245 

276 

307 

337 

5i/2X5i/2 

8.59 

52 

103 

155 

206 

258 

309 

361 

412 

464 

515 

567 

X4 

6.25 

37 

75 

112 

150 

188 

225 

262 

300 

337 

375 

412 

5  X5 

7.10 

43 

85 

128 

170 

213 

256 

298 

341 

383 

426 

469 

X4 

5.68 

34 

68 

102 

136 

170 

205 

239 

273 

307 

341 

375 

41/2X41/2 

5.75 

35 

69 

104 

138 

173 

207 

242 

276 

311 

345 

380 

X4 

5.11 

31 

61 

92 

123 

153 

184 

215 

246 

276 

307 

338 

4  X4 

4.54 

27 

55 

82 

109 

136 

164 

191 

218 

246 

272 

300 

X31/2 

3.97 

24 

48 

72 

96 

119 

143 

167 

181 

215 

238 

262 

X3 

3.40 

20 

41 

61 

82 

102 

122 

143 

163 

184 

204 

224 

31/2X31/2 

3.48 

21 

42 

63 

84 

104 

125 

146 

167 

188 

209 

230 

X3 

2.98 

18 

36 

54 

72 

89 

107 

'  125 

143 

161 

179 

197 

3  X3 

2.56 

15 

31 

46 

61 

77 

92 

108 

123 

138 

154 

169 

ROOFING   MATERIALS   AND   ROOF   CONSTRUCTION.     191 


ROOFING  MATERIALS  AND  ROOF  CONSTRUCTION. 

Approximate  Weight  of  Roofing  Materials. 

(American  Sheet  &  Tin  Plate  Co.) 


Material. 


Lb.  per 

sq.  ft. 


Corrugated  galvanized  iron,  No.  20,  unbearded 

Copper,  16  oz.  standing  seam .  .  .  .  , 

Felt  and  asphalt,  without  sheathing 

Glass,  i/s  in.  thick 

Hemlock  sheathing,  1  in.  thick 

Lead,  about  l/s  in.  thick 

Lath  and  plaster  ceiling  (ordinary) 

Mackite,  1  in.  thick,  with  plaster 

Neponset  roofing,  felt,  2  layers 

Spruce  sheathing,  1  in.  thick 

Slate,  3/i6  in.  thick,  3  in.  double  lap 

Slate,  l/s  in.  thick,  3  in.  double  lap 

Shingles,  6  in.  X  18  in.,  1/3  to  weather 

Skylight  of  glass,  3/ie  to  1/2  in.,  including  frame 

Slag  roof,  4-ply 

Terne  plate,  1C,  without  sheathing 

Terne  plate,  IX,  without  sheathing 

Tiles  (plain),  10  1/2  in.  X  6  1/4  in.  X  5/8  in.  -  5  1/4  in.  to  weather . 

Tiles  (Spanish),  14  1/2  in.  X  10  l/2  in.-  7  1/4  in.  to  weather 

White  pine  sheathing,  1  in.  thick 

Yellow  pine  sheathing,  1  in.  thick 


21/4 

,./< 
<»/< 

6  to  8 
6  to  8 
10 

1/2 
21/2 
63/4 

4l/2 

4  to  10 
4 
1/2 

5/8 
18 

81/2 
21/2 


Snow  and  Wind  Loads  on  Roofs. 

In  designing  roofs,  in  addition  to  the  weight  of  roofing  material  to 
be  supported,  recognition  must  be  given  to  possible  snow  and  wind  loads. 

In  snowy  localities  the  minimum  snow  load  per  horizontal  sq.  ft.  of 
roof  should  be  considered  as  25  Ib.  for  slopes  up  to  20  degrees.  For 
each  degree  increase  in  slope  up  to  45  degrees,  this  load  may  be  reduced 
1  Ib.  Above  45-degree  slope  no  snow  load  need  be  considered.  In 
especially  severe  climates  these  allowances  should  be  increased  in  ac- 
cordance with  actual  conditions. 

The  wind  load  is  the  pressure  normal  to  the  surface  of  the  roof  pro- 
duced by  a  wind  blowing  horizontally.  The  wind  pressure  against  a 
vertical  plane  as  determined  by  the  U.  S.  Signal  Service  at  Mt.  Wash- 
ington, N.  H.,  is  for  various  velocities  of  wind: 

Velocity,  miles  per  hr ..  10     20     30     40     50       60       80      100 

Pressure,  Ib.  per  sq.  ft 0.4    1.6    3.6    6.4    10.0    14.4    25.6    40.0 

The  pressure  on  a  flat  surface  is  twice  that  on  a  cylindrical  surface 
of  the  same  projected  area.  For  further  information  regarding  wind 
pressure,  see  page  626.  As  the  slope  of  the  roof  increases,  the  greater 
becomes  the  wind  pressure  on  it.  The  pressure  normal  to  the  surface 
of  roofs  of  different  slopes  exerted  by  a  wind  velocity  of  100  miles  per 
hour  (40  Ib.  per  sq.  ft.  on  a  vertical  plane)  is 

Rise,  in.  per  ft. .       4  6  8  12  16  18  24 

Angle  with 

horizontal.  . 
Pitch  (Rise  -j- 

Span) 1/6  1/4  V3  1/2  V3  V4  1 

Wind  pressure. .    16.8        23.7        29.1        36.1        38.7        39.3        40.0 

Roof  Construction.  (N.  G.  Taylor  Co.,  Philadelphia.) — Roofs  with 
less  than  1/3  pitch  are  made  with  flat  seams,  and  should  preferably  be 
covered  with  14  X  20  in.  sheets,  rather  than  with  20  X  28-in.  sheets,  as 
the  larger  number  of  seams  tend  to  stiffen  the  surface  and  prevent 
buckles.  For  a  flat  seam  roof  the  edges  of  the  sheets  are  turned  1/2  in., 
locked  together  and  soldered.  The  sheets  are  fastened  to  the  sheath.- 


.  18°  26'  26°  34'  33°  41'  45°  0'     53°  8'     56°  19'  63°  26' 


192 


MATERIALS, 


ing  boards  by  cleats  8  in.  apart  and  locked  in  the  seams.  Two  1-in 
barbed  and  tinned  nails  are  driven  in  each  cleat.  Steep  tin  roofs 
should  be  made  with  standing  seams  and  from  28  X  20-in.  sheets.  The 
sheets  are  first  single  or  double  seamed  and  soldered  together  in  a  long 
strip  reaching  from  eave  to  ridge.  The  sloping  seams  are  composed 
of  two  "upstands"  interlocked  at  the  upper  edge  and  held  to  the  sheath- 
ing boards  by  cleats.  No  solder  is  used  in  standing  seams  as  a  rule 
In  soldering  tin  roofs,  only  a  good  rosin  flux  should  be  used.  The  use 
of  acid  must  be  carefully  avoided. 

ttoof  Paints. — The  American  Sheet  and  Tin  Plate  Co.  recommends 
for  painting  metal  work  and  tin  roofs  metallic  brown,  Venetian  red,  or 
red  oxide  paint,  ground  in  pure  linseed  oil.  The  paint  should  be 
rubbed  well  in,  and  should  not  be  spread  thin.  See  also  Preservative 
Coatings,  page  471. 

Tin  Plates  are  made  of  soft  sheet  steel  coated  with  tin,  and  are 
called  in  the  trade  "coke"  or  "charcoal"  plates  according  to  the  weight 
of  coating.  These  terms  have  survived  from  the  time  when  the  highest 
quality  of  plate  was  made  from  charcoal-iron,  while  the  lower  grades 
were  made  from  coke-iron.  Consequently,  plates  to-day  with  the 
lighter  coatings  are  known  as  coke-plates,  and  are  used  for  tin  cans,  etc. 
The  various  grades  of  charcoal-plates  are  designated  by  the  letters  A  to 
AAAAA,  the  latter  having  the  heaviest  coating  and  the  highest  polish. 
There  is  one  other  brand  made  with  a  heavier  coating  than  5A,  which  is 
especially  adapted  for  nickel-plating.  The  unit  9f  value  and  measure- 
ment of  tin  plates  is  the  "base-box,"  which  will  hold  112  sheets  of 
14  X  20  in.  plate,  or  31360  sq.  in.  of  any  size.  Plates  lighter  than  65  Ib. 
per  base  box  (No.  36  gage)  are  known  as  taggers  tin. 

Weights  of  Standard  Galvanized  Sheets. 

(American  Sheet  &  Tin  Plate  Co.) 


1 

o 

M 

$£ 

&B 

d* 

| 

d 

O 

1* 

§£ 

fc^ 

5 

1 
O 

&£ 
$$ 

I* 

a* 

O 

M 

«s 

I* 
tf 

8 
9 
10 
11 
12 
13 
14 

112.5 
102.5 
92.5 
82.5 
72.5 
62.5 
52.5 

7.031 
6.406 
5.781 
5.156 
4.531 
3.906 
3.281 

15 
16 
17 
18 
19 
20 
21 

47.5 
42.5 
38.5 
34.5 
30.5 
26.5 
24.5 

2.969 
2.656 
2.406 
2.156 
1.906 
1.656 
1.531 

22 
23 
24 
25 
26 
27 
28 

22.5 
20.5 
18.5 
16.5 
14.5 
13.5 
12.5 

1.406 
1.281 
1.156 
1.031 
0.906 
.844 
.781 

29 
30 
31 
32 
33 
34 

11.5 
10.5 
9.5 
9.0 
8.5 
8.0 

0.719 
.656 
.594 
.563 
.531 
.500 

Standard  Weights  and  Gages  of  Tin  Plate. 

(American  Sheet  &  Tin  Plate  Co.,  Pittsburgh.) 


II 

Nearest 
Wire 
Gage  No. 

cr 

O3 

I* 
^ 

S.g 

PQo 

^(N 

°x 

F£ 

<u  £ 

T3   S 

^ 

Nearest 
Wire 
Gage  No. 

a1 

CO 

fe£ 
a  ... 

^ 

gfl 

«i 
°X 

^a 

100 
107 
118 
135 
128 
139 
155 
148 
175 

•§§ 

g 

Nearest 
Wire 
Gage  No. 

O" 

CQ   . 

§53 
P.  r 

^ 

Sg 

§ 

F* 

55  Ib. 
60 
65 
70 
75 
80 
85 
90 
95 

38 
37 
36 
35 
34 
33 
32 
31 
31 

0.252 
.275 
.298 
.321 
.344 
.367 
.390 
.413 
.436 

55 

60 
65 
70 
75 
80 
85 
90 
95 

lOOlb. 
1C 
1181b. 
IX 
IXL 
DC 
2X 
2XL 
3X 

30V2 
30 
29 
28 
28 
28 
27 
27 
26 

0.459 
.491 
.542 
.619 
.588 
.638 
.711 
.679 
.803 

3XL 
DX 
4X 
4XL 
D2X 
D3X 
D4X 

26 
26 
25 
25 
24 
23 
22 

0.771 
.826 
.895 
.863 
.964 
1.102 
1.239 

168 
180 
195 
188 
210 
240 
270 

TIN   AND   TERNE   PLATES. 


193 


Sizes  and  Net  Weight  per  Box  of  100  Ib.  (0.459  Ib.  per  sq.  ft.) 
Tin  Plates. 


Size  of 
Sheets. 

Sheets 
per 
Box. 

Weight 
per 
Box. 

Size  of 
Sheets. 

Sheets 
per 
Box. 

Weight 
per 
Box. 

Size  of 
Sheets. 

Sheets 
per 
Box. 

Weight 
per 
Box. 

10       X14 

225 

100 

15X15 

225 

161 

14       X31 

112 

155 

14       X20 

112 

100 

16X16 

225 

183 

111/4X223/4 

112 

91 

20       X28 

112 

200 

17X17 

225 

206 

131/4x173/4 

112 

84 

10      X20 

225 

143 

18X18 

112 

116 

131/4X191/4 

112 

91 

11       X22 

225 

172 

19X19 

112 

129 

131/2x191/2 

112 

94 

11i/2'X23 

225 

189 

20X20 

112 

143 

131/2x193/4 

112 

95 

12       X12 

225 

103 

21X21 

112 

158 

14       Xl83/4 

124 

103 

12       X24 

112 

103 

22X22 

112 

172 

14       X19V4 

120 

103 

13       X13 

225 

121 

23X23 

112 

189 

14       X21 

112 

105 

13       X26 

H2 

121 

24X24 

112 

204 

14       X22 

.112 

110 

14       X14 

225 

140 

26X26 

112 

241 

14       X221/4 

112 

111 

14       X28 

112 

140 

16X20 

112 

114 

15V2X23 

112 

127 

For  weight  per  box  of  other  than  100-lb.  plates  multiply  by  the 
figures  in  the  column  "Weight  per  Box"  in  the  preceding  table,  and 
divide  by  100.  Thus  for  IX  plates  20  X  28  in.,  200  X  135  +  100  =  270. 

Sheets  Required  for  Tin  Roofing. 

(American  Sheet  &  Tin  Plate  Co.,  1914.) 


Sheets 

Sheets 

Sheets 

Sheets 

Sheets 

e 

Required. 

£ 

Required 

.j 

Required. 

+1 

Required. 

42 

Required. 

S_4 

R 

g^_. 

, 

£_; 

w 

g^_. 

£H 

Is 

CO 

1^ 

s^ 

cr 

CO 

1^ 

2^ 

s 

1 

S&H 

1 

§^ 

s^ 

D* 
CO 

1^ 

*o 

^X 

^* 

"o 

^  V 

T3  V 

*o 

co^ 

'O  V 

"o 

^  V 

3* 

"8 

co§ 

^ 

1 

£~ 

CO 

6 
fc 

"S 
ST 

S3  •*• 

CO^ 

6 

I- 

co^ 

6 
fc 

!- 

Is 

CO 

1 

ICTJ- 

Is 

CO 

100 

59 

31 

280 

164 

86 

460 

269 

141 

640 

374 

197 

820 

479 

252 

110 

65 

34 

290 

170 

89 

470 

275 

144 

650 

379 

200 

830 

484 

255 

120 

70 

37 

300 

175 

92 

480 

280 

148 

660 

385 

203 

840 

490 

258 

130 

76 

40 

310 

181 

95 

490 

286 

151 

670 

391 

206 

850 

496 

26! 

140 

82 

43 

320 

187 

99 

500 

292 

154 

680 

397 

209 

860 

502 

264 

150 

88 

46 

330 

193 

102 

510 

298 

157 

690 

403 

212 

870 

508 

267 

160 

94 

50 

340 

199 

105 

520 

304 

160 

700 

409 

215 

880 

514 

270 

170 

100 

53 

350 

205 

108 

530 

309 

163 

710 

414 

218 

890 

519 

273 

180 

105 

56 

360 

210 

540 

315 

166 

720 

420 

221 

900 

525 

276 

190 

111 

59 

370 

216 

114 

550 

321 

169 

730 

426 

224 

910 

531 

279 

200 

117 

62 

380 

222 

'117 

560 

327 

172 

740 

432 

227 

920 

537 

282 

210 

123 

65 

390 

228 

120 

570 

333 

175 

750 

438 

230 

930 

543 

285 

220 

129 

68 

400 

234 

123 

580 

339 

178 

760 

444 

233 

940 

549 

288 

230 

135 

71 

410 

240 

126 

590 

344 

181 

770 

449 

236 

950 

554 

291 

240 

140 

74 

420 

245 

129 

600 

350 

184 

780 

455 

239 

960 

560 

295 

250 

146 

77 

430 

251 

132 

610 

356 

187 

790 

461 

243 

970 

566 

298 

260 

152 

80 

440 

257 

135 

620 

362 

190 

800 

467 

246 

980 

572 

301 

270 

158 

83 

450 

263 

138 

630 

368 

194 

810 

473 

249 

990 

578 

304 

Terne  Plates,  or  Roofing  Tin,  are  coated  with  an  alloy  of  tin  and  lead. 
In  the  "U.  S.  Eagle,  N.M."  brand  the  alloy  is  32%  tin,  68%  lead. 
The  weight  per  112  sheets  of  this  brand  before  and  after  coating  is  as 
follows: 

1C  14  X  20  1C  20  X  28  IX  14  X  20  IX  20  X  28 
Black  plates ...  95  to  100  Ib.  190  to  200  Ib.  125  to  130  Ib.  250  to  260  Ib. 
After  coating. .  .  115  to  120  230  to  240  145  to  150  290  to  300 

Terne  plates  are  made  in  two  thicknesses:  1C,  in  which  the  iron  body 
weighs  about  50  Ib.  per  100  sq.  ft.,  and  IX,  in  which  it  weighs  62  1/2  Ib. 
per  1.00  sq.  ft.  The  1C  grade  is  preferred  for  roofing,  wnile  the  !?C 


194 


MATERIALS. 


grade  is  used  for  spouts,  valleys,  gutters,  and  flashings.  The  standard 
weight  of  14  X  20  in.  1C  plates  is  107  Ib.  per  base-box,  and  of  14  x  20- 
in.  IX  plate  135  Ib. 

Long  terne  sheets  are  made  in'gages,  Nos.  14  to  32,  from  10  to  40  in. 
wide  and  up  to  120  in.  long.  They  are  made  in  five  grades  with  coat- 
ings of  8,  12,  15,  20,  and  25  Ib. 

A  box  of  112  sheets  14  X  20  in.  will  cover  approximately  192  sq.  ft. 
of  roof,  flat  seam,  or  583  sheets  1000  sq.  ft.  For  standing  seam  roofing 
a  sheet  20  X  28  in.  will  cover  475  sq.  in.,  or  303  sheets  1000  sq.  ft.  A 
box  of  112  sheets  20  X  28  in.  will  cover  approximately  366  sq.  ft. 

The  common  sizes  of  tin  plates  are  10  X  14  in.  and  multiples  of  that 
measure.  The  sizes  most  generally  used  are  14  x  20  and  20  X  28  in. 

Specifications  for  Tin  and  Terne  Plate.     (Penna.  R.R.,  1903.) 


Material  Desired. 

Rejected  if  less  than 

Tin 
Plate. 

No.  1 
Terne. 

No.  2 
Terne. 

Tin 
Plate. 

No.  1 

Terne. 

No.  2 
Terne. 

Coating: 
Tin,  per  cent  

100 
0 
0.023 

0.496 
.625 
.716 
.808 
.900 

26 
74 
0.046 

0.519 
.648 
.739 
.831 
.923 

16 
84 
0.023 

0.496 
.625 
.716 
.808 
.900 

Lead,  per  cent  

Amount  per  sq.  ft.,  Ib.  . 
Weight,  Ib.  per  sq.  ft.  of 
Grade  1C... 

0.0183 

0.468 
.590 
.676 
.763 
.850 

0.0413 

0.490 
.612 
.699 
.787 
.874 

0.083 

0.468 
.590 
.676 
.763 
.850 

Grade  IX  ... 

Grade  IXX  

Grade  IXXX  .  . 

Grade  IXXXX  

Each  sheet  in  a  shipment  of  tin  or  terne  plate  must  (1)  be  cut  as 
nearly  exact  to  size  ordered  as  possible;  (2)  must  be  rectangular,  flat, 
and  free  from  flaws;  (3)  must  double  seam  successfully  under  reason- 
able treatment;  (4)  must  show  a  smooth  edge  with  no  sign  of  fracture 
when  bent  through  an  angle  of  180  degrees  and  flattened  down  with  a 
wooden  mallet ;  (5)  must  be  so  nearly  like  every  other  sheet  in  the  ship- 
ment, both  in  thickness  and  in  uniformity  and  amount  of  coating,  that 
110  difficulty  will  arise  in  the  shops  due  to  varying  thickness  of  sheets. 

Corrugated  Sheets. — Weight  per  100  Sq.  Ft.,  Lb. 

(American  Sheet  &  Tin  Plate  Co.,  Pittsburgh,  1914.) 


Corruga- 
tions. 

5/8  in. 

1V4  in. 

2  in. 

2  i/2  in.* 
26  in. 
wide. 

2  1/2  in.f 
27  i/2  in. 
wide. 

3  in. 

5  in. 

U.  S.  Std. 
Sheet 
Metal 
Gage. 

"8 
a 

& 

jh 

eft  N 

o 

1 

a 
'3 
ft 

i 

11 

o'rt 

! 

ia 

PH 

li 

O'~ 

I 
1 

jb 

5* 

1 

& 

ji 

1 

1 
PH 

li 

0'" 

1 
'<« 

PH 

"68 
75 
81 
95 
108 
122 
135 
148 
162 
215 
269 
336 
470 

h 

73  N 

0'~ 

~~77 

84 
91 
97 
111 
124 
137 
151 
164 
178 
231 
285 
352 
486 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
18 
16 
14 
12 
10 

"i\ 

78 
85 
99 
113 

81 
88 
95 
102 
116 
130 

"7\ 
78 
85 
99 
113 
127 
141 
155 
169 

81 
88 
95 
102 
116 
130 
144 
158 
172 
186 

"68 
75 
82 
95 
109 
122 
136 
149 
163 
216 
270 

77 
84 
91 
98 
111 
125 
138 
151 
165 
178 
232 
286 

"68 
75 
82 
95 
109 
122 
136 
149 
163 
216 
270 
338 
472 
607 

77 
84 
91 
98 
111 
125 
138 
151 
165 
178 
232 
286 
353 
488 
623 

69 
76 
83 
97 
110 
124 
137 
151 
165 
219 
274 
342 
478 
615 

78 
85 
92 
99 
113 
126 
140 
153 
167 
181 
235 
290 
358 
494 
631 

"68 
75 
82 
95 
109 
122 
136 
149 
163 
216 
270 
338 
472 

77 
84 
91 
98 
111 
125 
138 
151 
165 
178 
232 
286 
353 
488 

.... 

.... 

*  Siding.        t  Roofing, 


SLATE. 


195 


Covering  width  of  plates,  lapped  one  corrugation.  24  in.  Standard 
lengths,  5,  6,  7,  8,  9,  and  10  ft.;  maximum  length,  12  ft. 

Ordinary  corrugated  sheets  should  have  a  lap  of  1 1/2  or  2  corrugations 
side-lap  for  roofing  in  order  to  secure  water-tight  side  seams ;  if  the  roof 
is  rather  steep  1 1/2  corrugations  will  answer.  Some  manufacturers 
make  a  special  high-edge  corrugation  on  sides  of  sheets,  and  thereby  are 
enabled  to  secure  a  water-proof  side-lap  with  one  corrugation  only,  thus 
saving  from  6%  to  12%  of  material  to  cover  a  given  area. 

No.  28  gage  corrugated  iron  is  generally  used  for  applying  to  wooden 
buildings;  but  for  applying  to  iron  framework  No.  24  gage  or  heavier 
should  be  adopted. 

Galvanizing  sheet  iron  adds  about  21/2  oz.  to  its  weight  per  square 
foot. 

Slate. 

Slate  in  roofs  is  measured  by  the  square,  1  square  being  equal  to  100 
superficial  square  feet.  In  measuring,  the  width  of  the  eaves  is  allowed 
at  the  widest  part.  Hips,  valleys,  and  cuttings  are  measured  lineally 
and  6  in.  extra  is  allowed.  The  thickness  of  slate  for  roofing  varies 
usually  from  1/8  to  3/16  in.  The  weight  varies,  when  lapped,  from 
4  1/2  to  63/4  lb.  per  sq.  ft.  The  laps  range  from  2  to  4  in.,  3  in.  being 
the  standard.  As"  slate  is  usually  laid,  the  number  of  square  feet  of  roof 
covered  by  one  slate  is  w  (I  —  3)  -~  288,  w  and  I  being  the  width  and 
length  respectively  of  the  slate  in  inches. 

Number  and  Superficial  Area  of  Slate  for  One  Square  of  Roof. 


Size, 
In. 

No. 
per 
Sq. 

Area, 

£ 

Size, 
In. 

No. 

K 

Area, 
Sq. 
Ft. 

Size, 
In. 

No. 

I? 

Area, 

1?: 

Size, 
In. 

No. 

per 
Sq. 

Area, 
Sq. 
Ft. 

6X12 
7X12 

533 
457 

267 

10X14 
8X16 

261 

277 

246 

10X20 
11  X20 

169 
154 

235 

12X24 
14X24 

114 
98 

228 

8X12 
9X12 
7X14 
8X14 

400 
355 
374 
327 

'254' 

9X16 
10X16 
9X18 
10X18 

246 
221 
213 
192 

'240' 

12X20 
14X20 
16X20 
12  X22 

141 
121 
137 
126 

23J 

16X24 
14X26 
16X26 

86 
89 
78 

'225' 

9X14 

291 

12X18 

160 

240 

14X22 

108 

Weight  of  Slate,  in  Pounds,  for  One  Square  of  Roof. 

(1  cu.  ft.  slate   =  175  lb.) 


Length 
of 
Slate,  In. 

Thickness  of  Slate,  Inch. 

Vs 

3/16 

V4 

3/8 

Va 

5/8 

3/4 

1 

!>4' 

16 
18 
20 
22 
24 
26 

483 
460 
445 
434 
425 
418 
412 
407 

724 
688 
667 
650 
637 
626 
617 
610 

967 
920 
890 
869 
851 
836 
825 
815 

1450 
1379 
1336 
1303 
1276 
1254 
1238 
1222 

1936 
1842 
1784 
1740 
1704 
1675 
1653 
1631 

2419 
2301 
2229 
2174 
2129 
2093 
2066 
2039 

2902 
2760 
2670 
2607 
2553 
2508 
2478 
2445 

3872 
3683 
3567 
3480 
3408 
3350 
3306 
3263 

Corrugated  Arches. 

For  corrugated  curved  sheets  for  floor  and  ceiling  construction  in 
fireproof  buildings,  No.  16,  18,  or  20  gage  iron  is  commonly  used,  and 
sheets  may  be  curved  from  4  to  10  in.  rise — the  higher  the  rise  the 
stronger  the  arch.  By  a  series  of  tests  it  has  been  demonstrated  that 
corrugated  arches  give  the  most  satisfactory  results  with  a  base  length 
not  exceeding  6  ft.,  and  5  ft.  or  even  less  is  preferable  where  great 
strength  is  required.  These  corrugated  arches  are  made  with  1 1/4  X  3/8, 


196 


MATERIALS. 


2  1/2  X  1/2,  3  X  3/4,  and  5  X  Vs  in.  corrugations,  and  in  the  same  width 
of  sheet  as  above  mentioned. 

Terra-Cotta. 

Porous  terra-cotta  roofing  3  in.  thick  weighs  16  Ib.  per  square  foot  and 
2  in.  thick  12  Ib.  per  square  foot. 

Ceiling  made  of  the  same  material  2  in.  thick  weighs  11  Ib.  per  square 
foot. 

Tiles. 

Flat  tiles  61/4  X  101/2  X  5/8  in.  weigh  from  1480  to  1850  Ib.  per  square 
of  roof  (100  square  feet),  the  lap  being  one-half  the  length  of  the  tile. 

Tiles  with  grooves  and  fillets  weigh  from  740  to  925  Ib.  per  square  of 
roof. 

Pan-tiles  141/2  X  101/2  laid  10  in.  to  the  weather  weigh  850  Ib.  per 
square. 

Pine  Shingles. 

The  figures  below  give  the  weight  of  shingles  required  to  cover  one 
square  of  a  common  gable  roof.  For  hip  roofs  add  5  per  cent. 

Inches  exposed  to  weather.  .  .............     4      41/2      5      51/2      6 

No.  of  shingles  per  square  of  roof  .........   900     800     720     655     600 

Weight  of  shingles  per  square,  Ib  .........   216     192     173     157     144 

Skylight  Glass  Required  for  One  Square  of  Roof. 

Dimensions,  in  ...............  12  X  48  15  X  60  20  X  100  94  X  156 

Thickness,  in  ........  ,  ........        3/16  i/4             3/8  l/2 

Area,  sq.  ft  ..................  3.997  6.246        13.880  101.768 

Weight  per  square,  Ib  .........        250  350            500  700 

No  allowance  has  been  made  in  the  above  figures  for  lap.  If  ordinary 
window-glass  is  used,  single  thick  glass  (about  Vie  inch)  will  weigh  about 
82  Ib.  per  square,  and  double  thick  glass  (about  i/s  inch)  will  weigh 
about  164  Ib.  per  square,  no  allowance  being  made  for  lap.  A  box  of 
ordinary  window-glass  contains  as  nearly  50  square  feet  as  the  size  of 
the  panes  will  admit.  Panes  of  any  size  are  made  to  order  by  the 
manufacturers,  but  a  great  variety  of  sizes  are  usually  kept  in  stock, 
ranging  from  6X8  inches  to  36  X  60  inches. 

THICKNESS  OF  CAST-IRON  WATER-PIPES. 

P.  H.  Baermann,  in  a  paper  read  before  the  Engineers'  Club  of  Phila- 
delphia in  1882,  gave  twenty  different  formulae  for  determining  the 
thickness  of  cast-iron  pipes  under  pressure.  The  formulae  are  of  three 
classes: 

1.  Depending  upon  the  diameter  only. 

2.  Those  depending  upon  the   diameter  and  head  and  which  add  a 
constant. 

3.  Those  depending  upon  the  diameter  and  head  contain  an  additive 
or  subtractive  term  depending  upon  the  diameter,  and  add  a  constant. 

The  more  modern  formulae  are  of  the  third  class,  and  are  as  follows: 

t  =  0.00008/id  +  O.Old  +  0.36  ................  Shedd,  No.  1. 

t  =  0.00006/id  +  0.0133d  +  0.296  .............  Warren  Foundry,  No.  2. 

t  =  0.000058M  +  0.0152d  4-  0.312  ............  Francis,  No.  3. 

t  =  0.000048/id  4-  0.013d  +  0.32  ..............  Dupuit,  No.  4. 

t  =  0.00004/id  4-  0.1    Vd~4-  0.15  .....  .........  Box,  No.  5. 

t  =  0.000135/id  4-  0.4  -  0.001  Id  ..............  Whitman,  No.  6. 

t  =  0.00006  (h  4-  230)  d  4-  0.333  -  0.0033d  ......  Fanning,  No.  7. 

t  =  O.OOOlSftd  +  0.25  -  0.0052d  ...............  Meggs,  No.  8. 

In  which  t  =  thickness  in  inches,  h  =  head  in  feet,  d  =  diameter  in 
inches.  For  h  =  100  ft.,  and  d  =  10  in.,  formulas  Nos.  1  to  7  inclusive 
give  to  from  0.49  to  0.54  in.,  but  No.  8  gives  only  0.35  in.  Fanning's 
formula,  now  (1908)  in  most  common  use,  gives  0.50  in. 

Rankine  (Civil  Engineering},  p.  721,  says:  "Cast-iron  pipes  should  be 
made  of  a  soft  and  tough  quality  of  iron.  Great  attention  should  be  paid 


THICKNESS  OF  CAST-IRON  WATEB-HPES.        1Q7 


to  molding  them  C9rrectly,  so  that  the  thickness  may  be  exactly  uniform 
all  round.  Each  pipe  should  be  tested  for  air-bubbles  and  flaws  by  ring- 
ing it  with  a  hammer,  and  for  strength  by  exposing  it  to  double  the 
intended  greatest  working  pressure."  The  rule  for  computing  the  thick- 
ness of  a  pipe  to.  resist  a  given  working  pressure  is  t  =  rp/f,  where  r  is 
the  radius  in  inches,  p  the  pressure  in  pounds  per  square  inch,  and /the 
tensile  strength  of  the  iron  per  square  inch.  When  /  =  18,000,  and  a 
factor  of  safety  of  5  is  used,  the  above  expressed  in  terms  of  d  and  h 
becomes  t  =  0.5d  X  0.433/1  -T-  3600  =  0.00006d/i. 

"There  are  limitations,  however,  arising  from  difficulties  in  casting, 
and  by  the  strain  produced  by  shocks,  which  cause  the  thickness  to  be 
made  greater  than  that  given  by  the  above  formula."  (See  also  Burst- 
ing Strength  of  Cast-iron  Cylinders,  under  "Cast  Iron.") 

The  most  common  defect  of  cast-iron  pipes  is  due  to  the  "shifting  of 
the  core,"  which  causes  one  side  of  the  pipe  to  be  thinner  than  the  other. 
Unless  the  pipe  is  made  of  very  soft  iron  the  thin  side  is  apt  to  be  chilled 
in  casting,  causing  it  to  become  brittle  and  it  may  contain  blow-holes 
and  "  cold-shots."  This  defect  should  be  guarded  against  by  inspection 
of  every  pipe  for  uniformity  of  thickness. 

Standard  Thicknesses  and  Weights  of  Cast-iron  Pipe. 

(U.  S.  Cast  Iron  Pipe  &  Foundry  Co.,  1915.) 


:§  . 

Class  A. 

Class  B. 

Class  C. 

Class  D. 

100  Ft.  Head. 

200  Ft.  Head. 

300  Ft.  Head. 

400  Ft.  Head. 

££ 

43  Lb.  Pressure. 

86  Lb.  Pressure. 

130  Lb.  Pressure. 

1  73  Lb.  Pressure 

•rt  c 

.6  c3 

** 

Pounds  per 

AS 

Pounds  per 

%& 

Pounds  per 

%* 

Pounds  per 

|S 

£ 

•S         - 

Ft. 

L'gth. 

2  if 

El 

Ft. 

L'gth. 

PH  ® 

Ft. 

L'gth. 

g| 

Ft. 

Lgfch. 

3 

0.39 

14.5 

175 

0.42 

16.2 

194 

0.45 

17.1 

205 

0.48 

18.0 

216 

4 

.42 

20.0 

240 

.45 

21.7 

260 

.48 

23.3 

280 

.52 

25.0 

300 

6 

.44 

30.8 

370 

.48 

33.3 

400 

.51 

35.8 

430 

.55 

38.3 

460 

8 

.46 

42.9 

515 

.51 

47.5 

570 

.56 

52.1 

625 

.60 

55.8 

670 

10 

.50 

57.1 

685 

.57 

63.8 

765 

.62 

70.8 

850 

.68 

76.7 

920 

12 

.54 

72.5 

870 

.62 

82.1 

985 

.68 

91.7 

1100 

.75 

100.0 

1200 

14 

.57 

89.6 

1075 

.66 

102.5 

1230 

.74 

116.7 

1400 

.82 

129.2 

1550 

16 

.60 

108.3 

1300 

.70 

125.0 

1500 

.80 

143.8 

1725 

.89 

158.3 

1900 

18 

.64 

129.2 

1550 

.75!    150.0 

1800 

.87 

175.0 

2100 

.96 

191.7 

2300 

20 

.67 

150.0 

1800 

.80 

175.0 

2100 

.92 

208.3 

2500 

.03 

229.2 

2'/50 

24 

.76 

204.2 

2450 

.89 

233.3 

2800 

.04 

279.2 

3350 

.16 

306.7 

3680 

30 

.88 

291.7 

3500 

.03 

333.3 

4000 

.20 

400.0 

4800 

.37 

450.0 

5400 

36 

.99    391.7 

4700 

.15 

454.2 

5450 

.36 

545.8 

6550 

.58 

625.0 

7500 

42 

.10'    512.5 

6150 

.28 

591.7 

7100 

.54 

716.7 

8600 

.78 

825.0 

9900 

48 

.26!    666.7 

8000 

.42 

750.0 

9000 

.71 

908.3 

10900 

.96  1050.0 

12600 

54 

.35    800.0 

9600 

.55 

933.3 

11200 

.90 

1141.7 

13700 

2.23il341.7 

16100 

60 

.39    916.7 

11000 

.67 

1104.2 

13250 

2.00 

1341.7 

16100 

2.38 

1583.3 

19000 

72 

.62  1281.9 

15380 

.95 

1547.3 

18570 

2.39 

1904.3 

22850 

. 

84 

.72!  1635.8 

19630 

2.22 

2104.1 

25250 

The  above  weights  are  per  length  to  lay  12  feet,  including  standard 
sockets;  proportionate  allowance  to  be  made  for  any  variation. 

Weight  of  Underground  Pipes.  (Adopted  by  the  Natl.  Fire  Pro- 
tection Association,  1913.)  Weights  are  not  to  be  less  than  those 
specified  when  the  normal  pressures  do  not  exceed  125  Ib.  per  sq.  in. 
When  the  normal  pressures  are  in  excess  of  125  Ib.  heavier  pipes  should 
be  used.  The  weights  given  include  sockets. 

Pipe,  in. . ,  46  8  10         12  14  16 

Weights  per  foot,  Ib....   23     35.8     52.1     70.8     91.7     116.7     143.8 


198 


MATERIALS. 


Standard  Thicknesses  and  Weights  of  Cast  Iron  Pipe. 
For  Fire  Lines  and  High-Pressure  Service. 

(U.  S.  Cast  Iron  Pipe  &  Foundry  Co.,  1915.) 


Nominal  Inside 
Diam.,  In. 

Class  E. 
500  ft.  Head. 
217-lb.  Pressure. 

Class  F. 
600  ft.  Head. 
260-lb.  Pressure. 

Class  G. 
700  ft.  Head. 
3044b.  Pressure. 

Class  H. 
800  ft.  Head. 
347-lb.  Pressure. 

ft 

r  .    O> 

&  c 

Lb.  per 

A& 

_o    - 

Lb.  per 

if 

Hg 

Lb.  per 

ii 

r.    0> 
^    C 

Lb.  per 

Ft. 

Lgth. 

Ft. 

Lgth. 

Ft. 

Lgth. 

Ft. 

Lgth. 

6 
8 
10 
12 
14 
16 
18 
20 
24 
30 
36 

0.58 
.66 
.74 
.82 
.90 
.98 
.07 
.15 
.31 
.55 
.80 

42.5 
60.9 
86.9 
114.6 
145.6 
180.7 
221.8 
265.8 
359.1 
530.9 
738.1 

510 
731 

1043 
1375 
1747 
2168 
2662 
3190 
4309 
6371 
8857 

0.61 
.71 
.80 
.89 
.99 
1.08 
1.17 
1.27 
1.45 
1.73 
2.02 

44.3 
66.8 
92.8 
122.8 
158.8 
196.5 
239.3 
287.3 
392.3 
588.8 
821.0 

531 

802 
1114 
1474 
1905 
2358 
2872 
3448 
4707 
7065 
9852 

0.65 
.75 
.86 
.97 
.07 
.18 
.28 
.39 
.75 

48.1 
72.3 
101.4 
136.2 
175.1 
218.0 
268.2 
321.8 
479.8 

577 

868 
1217 
1634 
2101 
2616 
3218 
3862 
5758 

0.69 
.80 
.92 
.04 
.16 
.27 
.39 
.51 
.88 

50.5 
76.1 
107.3 
144.4 
187.5 
233.8 
287.8 
345.8 
510.6 

606 
913 
1288 
1733 
2250 
2805 
3453 
4149 
6127 

All  lengths  to  lay  12  ft.  Weights  are  approximate;  those  per  foot 
include  allowance  for  bell;  those  per  length  include  bell.  Propor- 
tionate allowance  is  to  be  made  for  variations  from  standard  length. 


Standard  and  Heavy  Cast  Iron  Bell  and  Spigot  Gas  Pipe. 
Weights  and  Dimensions. 

(U.  S.  Cast  Iron  Pipe  &  Foundry  Co.,  1914.) 


Actual  Out- 

Thickness, 

Dia.  of  Sock- 

A 

Weight  per 

Weight  per 

,—  fi 

side  Dia.,  In. 

In. 

ets,  In. 

*o  w" 

Foot,  Lb. 

Length,  Lb. 

•p 

$4 

£ 

'O    . 

CT3 

£ 

"O    . 

CTJ 

i 

O  4_> 

fl 

^  . 

fl"rt 

£ 

ro  . 

CTJ 

>» 

> 

§s 

11 

o> 

3a 

8 

la 

0) 

&o 

3s 

8 

OS  £ 

8 

fc 

&« 

w 

wrt 

w 

oa* 

w 

pW 

£* 

W 

02* 

w 

4 

4.80 

5.00 

0.40 

0.42 

5.80 

5.80    4.00 

19.33 

20.0 

232 

240 

6 

6.90,     7.10 

.43 

.47 

7.90 

7.90    4.00 

30.25 

32.8 

363 

394 

8 

9.05     9.05 

.45 

.49 

10.05 

9.85    4.00 

42.08 

45.3 

505 

544 

10 

11  .10i  11  .10 

.49 

.51 

12.10 

11  .90    4.00 

55.91 

58.7 

671 

703 

12 

13.20 

13.20 

.54 

.57 

14.20 

14.00    4.50 

73.83 

76.1 

886 

913 

16 

17.40 

17.40 

.62 

.65 

18.40 

18.40    4.50 

112.58 

117.2 

1351 

1406 

20 

21.60 

21  .60 

.6G 

.75 

22.85 

22.60    4.50 

153.83 

166.7    1846!  2000 

24 

25.80 

25.80 

.76 

.82 

27.05 

26.80    5.00 

206.41 

224.0    2477;  2688 

30 

31  .74 

32.00 

.85 

1  .00 

32.99 

33.00    5.00 

284.001323.9 

3408  3887 

36 

37.96 

38.30 

.95 

1.05 

39.21 

39.30    5.00 

379.25  442.7    4551    5312 

42 

44.20  44.50 

1  .07 

1  .26 

45.45 

45.50    5.00 

497.66  581  .3    5972  6975 

48 

50.50  50.80 

1.26 

1.38 

51  .75i   51  .80    5.00 

663.50!  739.  6    7962  8875 

The  Standard  pipe  listed  above  conforms  to  the  standard  adopted  by 
the  American  Gas  Institute  in  1911.  The  heavy  pipe  given  is  not  in- 
cluded in  the  A.  G.  I.  standards  but  is  used  by  many  gas  engineers  for 
service  under  paved  streets  with  heavy  traffic,  or  where  subsoil  condi- 
tions make  the  heavier  pipe  desirable.  Pipes  are  made  to  lay  12  ft. 
length.  Weights  per  foot  include  bell  and  bead.  Length  of  bead  = 
0.75  in.  for  4-  and  6-in.  pipe;  1.00  in.  for  8-  to48-in.  pipe.  Thickness  of 
bead  =  0.19  in.  for  4-  and  6-in.  pipe;  0.25-in.  for  8-  to  48-in.  pipe. 


LEAD   REQUIRED   FOR   CAST   IRON   PIPE  JOINTS.    199 


Standard  Flanged  Cast  Iron  Pipe  for  Gas. 

(United  Cast  Iron  Pipe  &  Foundry  Co.,  1914,  Am.  Gas.  Inst.  Std.,  1913.) 


Nomi- 
nal 

Thick- 
ness, 

Flange 
Diam., 

Flange 
Thick- 

Bolt 
Circle 

Bolts 

Wgt. 
Single 

Approx.  Wgt., 
Lb. 

Diam., 
In. 

In. 

In. 

ness, 

In.   ' 

No. 

Size, 
In. 

r  lange, 
Lib. 

Foot. 

Lgth. 

4 

0.40 

9.00 

0.72 

7.125 

~T~ 

0.625 

8.19 

18.62 

223 

6 

.43 

11.00 

.72 

9.125 

4 

.625 

10.46 

29.01 

348 

8 

.45 

13.00 

.75 

11  .125 

8 

.625 

12.65 

40.05 

481 

10 

.49 

16.00 

.86 

13.75 

8 

.625 

22.53!     54.71 

656 

12 

.54 

18.00 

.875 

15.75 

8 

.625 

25.96      71.34 

856 

16 

.62 

22.50 

.00 

20.00 

12 

.75 

39.68 

108.61 

1303 

20 

.68 

27.00 

.00 

24.50 

16 

.75 

51.10!    147.95 

1775 

24 

.76 

31  .00 

.125 

28.50 

16 

.75 

65.00 

197.38 

2369 

30 

.85 

37.50 

.25 

35.00 

20 

.875 

96.70 

273.45 

3281 

36 

.95 

44.00 

.375 

41  .25 

24 

.875 

132.26 

366.67 

4400 

42 

1.07 

50.75 

.56 

47.75 

28 

1.00 

186.83    483.48 

5802 

48 

1  .26 

57.00 

.75 

54.00 

32 

1  .00 

235.23 

647.36 

7768 

Pipe  is  made  in  12-ft.  lengths,  and  faced  Vie  in.  short  for  gaskets. 
Weight  per  foot  includes  flanges.  Flanges  are  Am.  Gas.  Inst.,  and  are 
different  from  the  "American  1914"  standard  for  water  and  steam  pipe. 
Pipes  heavier  than  above  may  be  made  by  reducing  internal  diameters. 

Threaded  Cast  Iron  Pipe. 

(U.  S.  Cast  Iron  Pipe  &  Foundry  Co.,  1914.) 


Nominal  diam.,  in  

3 

4 

6 

8 

10 

12 

Actual  outside  diam.,  in 

3.96 

5.00 

7.10 

9.30 

11   40 

13.50 

Thickness,  in.,  Class  B  

0.42 

0.45 

0.48 

0.51 

0.57 

0.62 

Wt.  per  foot,  Class  B    .  . 

14.6 

20.1 

31  .2 

43.9 

60.5 

78.9 

Thickness  in    Class  D 

0  48 

0  52 

0  55 

0  60 

0  68 

0  75 

Wt.  per  foot,  Class  D  

16.4 

22.8 

35.3 

51.2 

71.4 

93.7 

Quantity  of  Lead  Required  for  Cast  Iron  Pipe  Bell  and  Spigot  Joints. 

(U.  S.  Cast  Iron  Pipe  &  Foundry  Co.,  1914.) 


S 

Depth  of  Joint 

i 

Depth  of  Joint 

§  c 

2  In.    1  2  1/4  In.  |  2  1/2  In.  |  Solid. 

§« 

2  In. 

2  1/4  In.  |  2  1/2  In.  |  Solid. 

p 

Approx.  Weight  of  Lead  in  Joint.—  Lb. 

3~ 

Approx.  Weight  of  Lead  in  Joint.  —  Lb. 

3 

6.00 

6.50        7.00 

10.25 

74 

44.00     48.00      52.50 

95.00 

4 

7.50 

8.00 

8.75 

13.00 

30 

54.25 

59.50 

64.75 

117.50 

6 

10.25 

11.25 

12.25 

18.00 

36 

64.75 

71  .00 

77.25 

140.25 

8 

13.25 

14.50 

15.75 

23.00 

42 

75.25 

78.75 

85.50 

155.25 

10 

16.00 

17.50 

19.00 

31  .00 

48 

85.50 

94.00 

102.25 

202.25 

12 

19.00 

20.50 

22.50 

36.50 

54 

97.60 

107.10 

116.60 

238.60 

14 

22.00 

24.00 

26.00 

38.50 

60 

108.30 

118.80 

129.50 

255.50 

16 

30.00 

33.00 

35.75 

64.75 

72 

128.00 

140.50 

153.00 

302.50 

18 

33.80 

36.90 

40.00 

72.00 

84 

147.00 

161  .50 

175.60 

348.00 

20 

37.00 

40.50 

44.00 

80.00 

The  above  table  gives  the  calculated  weight  of  lead  required  for  pipe 
joints  both  with  and  without  gasket.  Weight  of  lead  taken  at  0.41 
Ib.  per  cu.  in.  Allowance  has  been  made  for  lead  to  project  beyond  the 
face  of  the  bell  for  calking.  Pipe  specifications  allow  lead  space  to  vary 
from  those  given  in  tables,  hence  the  weights  of  lead  may  vary  ap- 
proximately 11  to  16  per  cent  from  those  given  above, 


200 


MATERIALS 


Cast-iron  Pipe  Columns,  Weight  and  Safe  Loads,  Pounds. 

(U.  S.  Cast  Iron  Pipe  and  Foundry  Co.,  1914.) 


T  onrrfVi 

4-Inch  Pipe. 

6-Inch  Pipe. 

8-Inch  Pipe. 

1  0-Inch  Pipe. 

ijGngtn. 

Wgt. 

Load. 

Wgt. 

Load. 

Wgt. 

Load. 

Wgt. 

Load. 

6  ft.  0  in. 

160 

56070 

245 

100100 

359 

164410 

428 

224200 

6   6 

171 

54130 

262 

98310 

385 

162400 

464 

222300 

7   0 

183 

52190 

280 

96270 

410 

160350 

500 

220300 

7   6 

194 

50250 

298 

94100 

436 

1  58200 

535. 

218300 

8   0 

206 

48320 

316 

92040 

462 

1  56000 

571 

216200 

8   6 

217 

46440 

333 

89820 

487 

153600 

607 

213900 

9   0 

229 

44590 

351 

87620 

513 

1  5  1  200 

643 

211600 

9   6 

240 

42800 

368 

85450 

539 

148760 

678 

209300 

10   0 

251 

41050 

386 

83260 

564 

146260 

714 

206900 

10   6 

262 

39360 

404 

81040 

590 

143700 

750 

204500 

11   0 

274 

37730 

421 

78840 

615 

141160 

785 

202200 

11   6 

285 

36160 

439 

76700 

642 

138570 

821 

199800 

12   0 

297 

34670 

457 

74580 

667 

135920 

857 

197400 

12   6 

308 

33220 

474 

71600 

692 

133340 

893 

195000 

Base  and  Top  Castings. 

Ins.  square        10 

12 

14 

16 

Wt.,  Ibs.         65 

100 

145 

200 

Add  weight  of  base  and  top  castings  f9r  complete  weight  of  column. 
Loads  are  based  on  Gordon's  formula,  with  a  factor  of  safety  of  8. 


Weight  of  Open  End  Cast-Iron  Cylinders. 

Cast  iron  =  450  Ibs.  per  cubic  foot. 
Pounds  per  Lineal  Foot. 


Thick. 

Wgt. 

Thick. 

Wgt. 

Thick. 

Wgt. 

Thick. 

Wgt. 

Bore. 

of 
Metal. 

per 
Foot. 

Bore. 

of 
Metal. 

per 
Foot. 

Bore. 

of 

Metal. 

Foot. 

Bore. 

of 
Metal. 

per 
Foot. 

In. 

In. 

Lb. 

In. 

"in*. 

Lb. 

In. 

In. 

Lb. 

In. 

In. 

Lb. 

4 

3/8 

16.1 

11 

V2 

56.5 

17 

V8 

153.6 

24 

7/8 

213.7 

!/2 

22.1 

5/8 

71.3 

18 

5/8 

114.3 

1 

245.4 

5/8 

28.4 

3/4 

86.5 

3/4 

138.1 

26 

3/4 

197.0 

5 

3/8 

19.8 

12 

V2 

61.4 

7/8 

162.1 

7/8 

230.9 

1/2 

27.0 

5/8 

77.5 

19 

5/8 

120.4 

1 

265.1 

5/8 

34.4 

3/4 

93.9 

3/4 

145.4 

28 

3/4 

211.7 

6 

3/8 

23.5 

13 

V2 

66.3 

7/8 

170.7 

7/8 

248.1 

1/2 

31.9 

5/8 

83.6 

20 

5/8 

126.6 

1 

284.7 

5/8 

40.7 

3/4 

101.2 

3/4 

152.8 

30 

7/8 

265.2 

7 

3/8 

27.2 

14 

V2 

71.2 

7/8 

179.3 

304.3 

V2 

36.8 

5/8 

89.7 

21 

5/8 

132.7 

U/8 

343.7 

5/8 

46.8 

3/4 

108.6 

3/4 

160.1 

32 

7/8 

282.4 

8 

3/8 

30.8 

15 

5/8 

95.9 

7/8 

187.9 

1 

324.0 

1/2 

41.7 

3/4 

116.0 

22 

5/8 

138.8 

H/8 

365.8 

5/8 

52.9 

7/8 

136.4 

3/4 

167.5 

34 

7/8 

299.6 

9 

V2 

46.6 

16 

5/8 

102.0 

7/8 

196.5 

1 

343.7 

5/8 

59.1 

3/4 

123.3 

23 

3/4 

174  9 

H/8 

388.0 

3/4 

71.8 

7/8 

145.0 

7/8 

205.1 

36 

7/8 

316.6 

TO 

1/2 

51.5 

17 

5/8 

108.2 

235.6 

] 

363.1 

5/8 

65.2 

3/4 

130.7 

24 

8/4 

182.2 

H/8 

410.0 

3/4 

79.2 

The  weight  of  two  flanges  may  be  reckoned  =  weight  of  one  foot, 


WELDED   PIPE. 


201 


WROUGHT-IRON  (OR  STEEL)  WELDED  PIPE. 

For  discussion  of  the  Briggs  Standard  of  Wrought-iron  Pipe  Dimen- 
sions, see  Report  of  the  Committee  of  the  A.  S.  M.  E.  in  "Standard 
Pipe  and  Pipe  Threads,"  1886.  Trans.,  Vol.  VIII,  p.  29.  The  diameter 
of  the  bottom  of  the  thread  is  derived  from  the  formula  D  — 

(0.05D+  1.9)  x  i,  in  which  D  =  outside  diameter  of  the  tubes,  and  n 
the  number  of  threads  to  the  inch.  The  diameter  of  the  top  of  the 
thread  is  derived  from  the  formula  0.8  ^  X  2  +  d,  or  1.6  i  +  d,  in  which 

d  is  the  diameter  at  the  bottom  of  the  thread  at  the  end  of  the  pipe. 
The  sizes  for  the  diameters  at  the  bottom  and  top  of  the  thread  at  the 
end  of  the  pipe  are  as  follows : 

Standard  Pipe  Threads. 


Nom- 

m^3 

Diam. 

Diam. 

Nom- 

OS'S 

Diam. 

Diam. 

inal 

ijl 

of  Pipe 

of  Pipe 

inal 

6f  Pipe 

of  Pipe 

Size. 

Ex- 

at Root 

at  Top 

Size. 

Ex- 

QJ*""1 

at  Root 

at  Top 

ternal 

-C  ^ 

of 

of 

ternal 

S  cD 

of 

of 

Diam. 

Ha 

Thread. 

Thread. 

Diam. 

Hft 

Thread. 

Thread. 



1/8 

0.405 

27 

0.3339 

0.3931 

5 

5.563 

8 

5.2907 

5.4907 

1/4 

.540 

18 

.4329 

.5218 

6 

6.625 

8 

6.3460 

6.5460 

3/8 

.675 

18 

.5676 

.6565 

7 

7.625 

8 

7.3398 

7.5398 

1/2 

.840 

14 

.7013 

.8156 

8 

8.625 

8 

8.3336 

8.5336 

3/4 

1.050 

14 

.9105 

1.0248 

9 

9.625 

8 

9.3273 

9.5273 

1 

1.315 

111/2 

1  .  1  440 

1.2832 

10 

10.750 

8 

10.4453 

10.6453 

H/4 

1.660  111/2 

1  .4876 

1  .6267 

11 

11.750 

8 

11.4390 

11.6390 

H/2 

1.900;i1l/2 

1.7265 

1  .8657 

12 

12.750 

8 

12.4328 

12.6328 

2 

2.375I1U/2 

2.1995 

2.3386 

13 

14.000 

8 

13.6750 

13.8750 

21/2 

2.875      8 

2.6195 

2.8195 

14 

15.000 

8 

14.6688 

14.8688 

3 

3.500      8 

3.2406 

3.4406 

15 

16.000 

8 

15.6625 

15.8625 

31/2 

4.000      8 

3.7375 

3.9375 

170.D. 

17.000 

8 

16.6563 

16.8563 

4 

4.500      8 

4.2343 

4.  4343 

18O.D. 

18.000 

8 

17.6500 

17.8500 

4l/2 

5.000      8 

4.7313 

4.9313 

20  O.D. 

20.000 

8 

19.6375 

19.8375 

Tap  Drills  for  Pipe  Taps  (Briggs'  Standard) . 


Size  of 
Tap, 
In. 

Size  of 
Drill, 
In. 

Size  of 
Tap,  . 
In. 

Size  of 
Drill, 
In. 

Size  of 
Tap, 
In. 

Size  of 
Drill, 
In. 

Size  of 
Tap, 
In. 

Siz^of 
Drill, 
In. 

1/8 

V4 
3/8 
1/2 

21/64 
29/64 
19/32 
23/32 

3/4 

1  1/4 

1  V2 

,%« 

1    3/16 
1  15/32 
1  23/32 

2 

21/2 

31/2 

2  3/16 
2H/16 
3  5/16 
313/ifi 

4 

41/2 

6 

4  3/16 
4H/16 
5  1/4 
6  5/ii 

Having  the  taper,  length  of  full-threaded  portion,  and  the  sizes  at 
bottom  and  top  of  thread  at  the  end  of  the  pipe,  as  given  in  the  table, 
taps  and  dies  can  be  made  to  secure  these  points  correctly,  the  length 
of  the  imperfect  threaded  portions  on  the  pipe,  and  the  length  the  tap 
is  run  into  the  fittings  beyond  the  point  at  which  the  size  is  as  given,  or, 
in  other  words,  beyond  the  end  of  the  pipe,  having  no  effect  upon  the 
standard.  The  angle  of  the  thread  is  60°,  and  it  is  slightly  rounded  off 
at  top  and  bottom,  so  that,  instead  of  its  depth  being  0.866  its  pitch,  as 
is  the  case  with  a  full  V-thread,  it  is  4/5  the  pitch,  or  equal  to  0.8  -r-  n,  n 
being  the  number  of  threads  per  inch. 

Taper  of  conical  tube  ends,  1  in  32  to  axis  of  tube  =  %  inch  to  the 
foot  total  taper. 

The  thread  is  perfect  for  a  distance  (L)  from  the  end  of  the  pipe,  ex- 
pressed by  the  rule,  L  =  (0.8  D  +  4.8)  -j-n;  where  D  =  outside  diameter 


202 


MATERIALS. 


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WELDED  PIPE. 


203 


in  inches.  Then  come  two  threads,  perfect  at  the  root  or  bottom, 
but  imperfect  at  the  top,  and  then  come  three  or  four  threads  imperfect 
at  both  top  and  bottom.  These  last  do  not  enter  into  the  joint  at  all, 
but  are  incident  to  the  process  of  cutting  the  threads.  The  thickness 
of  the  pipe  under  the  root  of  the  thread  at  the  end  of  the  pipe  =  0.0175 
D  +  0.025  in. 

Briggs'  standard  gages  are  made  by  Pratt  &  Whitney  Co.,  Hartford, 
Conn. 

Standard  Welded  Pipe. — The  permissible  variation  in  weights  is  5% 
above  and  5%  below  those  given  in  the  table  on  the  opposite  page. 
Pipe  is  furnished  with  threads  and  couplings,  and  in  random  lengths 
unless  otherwise  ordered.  Weights  are  figured  on  the  basis  of  one 
cubic  inch  of  steel  weighing  0.2833  lb.,  and  the  weight  per  foot  with 
threads  and  couplings  is  based  on  a  length  of  20  feet,  including  the 
coupling,  but  shipping  lengths  of  small  sizes  will  usually  average  less 
than  20  feet.  Taper  of  threads  is  %  inch  diameter  per  foot  length  for 
all  sizes.  The  weight  of  water  contained  in  one  lineal  foot  is  based 
on  a  weight  of  62.425  pounds  per  cubic  foot,  which  is  the  weight  at  its 
maximum  density  (39.1°  F.) 

The  steel  used  for  lap-welded  pipe  has  the  following  average  analysis 
and  physical  properties: 

El.         Tens.    Elong. 
C        Mn     "'  S  P         Lim.         Str.     in  8  in. 

Bessemer 0.07     0.30     0.045     0.100     36,000     58,000     22% 

Open-hearth 0.09     0.40     0.035     0.025     33,000     53,000     25% 

Extra  Strong~Plpe.     (National  Tube  Company,  1915) 


. 

Length  of 

*  bc-w 

Diameter. 

i 

^J 

Circum- 
ference. 

Transverse  Area. 

Pipe  per 
Sq.  Foot. 

sfj 

. 

41 

|| 

1 

9 

|| 

ll 

t<  £ 

bi 

0) 

J 

J 

sUO 

53 

W 

& 

s 

^ 

& 

£ 

W 

^ 

^ 

Hw 

&W 

J 

In. 

In. 

In. 

Lb. 

In. 

In. 

Sq.In. 

Sq.  In. 

Sq.In 

Ft. 

Ft. 

Ft. 

Va 

0.405 

0.215 

.095 

0.314 

1.272 

0.675 

0.129 

0.036 

0.093 

9.431 

17.766 

3966.393 

1/4 

.540 

.302 

.119 

.535 

1.696 

.949 

.229 

.072 

.157 

7.073 

12.648 

2010.290 

3/8 

.675 

.423 

.126 

.738 

2.121 

1.329 

.358 

.141 

.217 

5.658 

9.030 

1024.689 

1/2 

.840 

.546 

.147 

1.087 

2.639 

1.715 

.554 

.234 

.320 

4.547 

6.995 

615.017 

'      3/4 

1.050 

.742 

.154 

1.473 

3.299 

2.331 

.866 

.433 

.433 

3.637 

5.147 

333.016 

1.315 

.957 

.179 

2.171 

4.131 

3.007 

1.358 

.719 

.639 

2.904 

3.991 

200.193 

1  1/4 

1.660 

1.278 

.191 

2.996 

5.215 

4.015 

2.164 

1.283 

.881 

2.301 

2.988 

112.256 

U/2 

1.900 

1.500 

.200 

3.631 

5.969 

4.712 

2.835 

1.767 

1.068 

2.010 

2.546 

81  .487 

2 

2.375 

1.939 

.218 

5.022 

7.461 

6.092 

4.430 

2.953 

1.477 

1.608 

1.969 

48.766 

21/2 

2.875 

2.323 

.276 

7.661 

9.032 

7.298 

6.492 

4.238 

2.254 

1.328 

1.644 

33.976 

3 

3.500 

2.900 

.300 

10.252 

10.996 

9.111 

9.621 

6.605 

3.016 

1.091 

1.317 

21.801 

31/2 

4.000 

3.364 

.318 

12.505 

12.566 

10.568 

12.566 

8.888 

3.678 

0.954 

1.135 

16.202 

4 

4.500 

3.826 

.337 

14.983 

14.137 

12.020 

15.904 

1  1  .497 

4.407 

.848 

0.998 

12.525 

41/2 

5.000 

4.290 

.355 

17.611 

15.708 

13.477 

19.635 

14.455 

5.180 

.763 

.890 

9.962 

5 

5.563 

4.813 

.375 

20.778 

17.477 

15.120 

24.306 

18.194 

6.112 

.686 

.793 

7.915 

6 

6.625 

5.761 

.432 

28.573 

20.813 

18.099 

34.472 

26.067 

8.405 

.576 

.663 

5.524 

7 

7.625 

6.625 

.500  38.048 

23.955 

20.813 

45.664 

34.472 

11.192 

.500 

.576 

4.177 

8 
9 

8.625 
9.625 

7.625 
8.625 

.500  43.388 
.50048.728 

27.096 
30.238 

23.955 
27.096 

58.426 
72.760 

45.663 
58.426 

12.763 
14.334 

.442 
.396 

.500 
.442 

3.154 
2.465 

10 

10.750 

9.750 

.500  54.735  33.772 

30.631 

90.763 

74.662 

16.101 

.355 

.391 

1.929 

11 

11.750 

10.750 

.50060.07536.914 

33.772 

108.434 

90.763 

17.671 

.325 

.355 

1.587 

12 

12.750  11.7*0 

.50065.415 

40.055 

36.914 

127.676 

108.434 

19.242 

.299 

.325 

1.328 

13 

14.  000  Si  3.  000 

.50072.091 

43.982 

40.841 

153.938 

132.732 

21.206 

.272 

.293 

1.085 

14 

15.000'  14.0001  .500  77.431  147.124 

43.982 

176.715 

153.93822.777 

.254 

.272 

0.935 

15 

16.000!  15.0001  .500182.771  50.265 

47.124 

201  .062 

176.715l24.347 

.238 

.254 

.815 

The  permissible  variation  in  weight  is  5%  above  and  5%  below. 
Furnished  with  plain  ends!  and  in  random  lengths  unless  otherwise 
ordered. 


204 


MATERIALS, 


Double  Extra  Strong  Pipe.    (National  Tube  Company,  1915.) 


Diameter. 

1 

li 

3 

Circum- 
ference. 

Transverse  Area. 

Length  of 
Pipe  per 
Sq.  Foot. 

J..& 

*-M 

1 

i* 

H 

i" 

Thickn 

s 

^ 

& 

11 

h-t 

il 

*  a 
H 

6-d 

£* 

HH 

-3 
% 

% 

-1 

a 

Int. 
Surface. 

O.SU 

43  C  « 

|35 
1 

In. 

In. 

In. 

Lb. 

In. 

In. 

Sq.In  Sq.In 

Sq.In 

Ft. 

Ft. 

Feet. 

1/2 

0.840 

0.2520.294 

1.714    2.639 

0.792 

0.554 

0.050 

0.504 

4.547 

15.157 

2887.165 

3/4 

1.050 

.4341   .308 

2.440    3.299 

1.363 

.8661     .148 

.718 

3.637 

8.801 

973.404 

1.315 

.599    .358 

3.659    4.131 

1.882 

1.358 

.282 

1.076 

2.904 

6.376 

510.998 

U/4 

1.660 

.896 

.382 

5.214    5.215 

2.815 

2.164      .630 

1.534 

2.301 

4.263 

228.379 

H/2 

1.900 

1.100 

.400    6.408;  5.969 

3.456 

2.835 

.950 

1.885 

2.010 

3.472 

151.526 

2 

2.375 

1.503 

.436    9.029   7.461 

4.722 

4.430 

1.774 

2.656 

1.608 

2.541 

81.162 

21/2 

2.875 

1.771 

.552.13.695!  9.032 

5.564 

6.492    2.464 

4.028 

1.328 

2.156 

58.457 

3 

3.500 

2.300    .60018.58310.996 

7.226 

9.621 

4.155 

5.4t>6 

1.091 

1.660 

34.659 

31/2 

4.000 

2.728!   .63622.850  12.  5b6 

8.570 

12.566    5.845 

6.721 

0.954 

1.400 

24.637 

4 

4.500 

3.152    .67427.541  14.137 

9.902 

15.904 

7.803 

8.101 

.848 

1.211 

18.454 

41/2 

5.000 

3.5801    .71032.53015.708 

1  1  .247 

19.635 

10.066 

9.569 

.763 

1.066 

14.306 

5 

5.563 

4.063    .75038.55217.477 

12.764 

24.306 

12.966 

1  1  .340 

.686 

0.940 

11.107 

6 

6.625 

4.897    .86453.16020.813 

15.384 

34.472 

18.835 

15.637 

.576 

.780 

7.646 

7 

7.625 

5.875 

.875  63.079  23.955 

18.457 

45.664 

27.109 

18.555 

.500 

.650 

5.312 

8 

8.625 

6.875    .875  72  424127.096  21  .598  58.426 

37.12221.304    .442      .555 

3.879 

The  permissible  variation  in  weight  is  10%  above  and  10%  below. 
Furnished  with  plain  ends  and  in  random  lengths  unless  otherwise 
ordered. 

Standard  Boiler  Tubes  and  Flues — Lap- Welded. 

(National  Tube  Company,  1915.) 


Diameter. 

1 

1 

Circum- 
ference. 

Transverse  Area. 

Length  of  Tube 
per  Sq.  Foot. 

|:K 

jN 

QjlS 

z* 

.5?  §3 

ii 

JB 

fa 

£-3 

3 

8 

§ 

fig 

*O  4J  O 

"5  §  S 

<§§ 

1° 

1 

f 

ia 

£j 

Ia 

3* 

B 

a 

l| 

*l 

J0' 

In. 

In. 

In. 

Lb. 

In. 

In. 

Sq.In. 

Sq.  In. 

Sq.In 

Ft. 

Ft. 

Ft. 

Ft. 

13/4 

1.560 

0.095 

1.679 

5.498 

4.901 

2.405 

1.911 

.494 

2.182 

2.448 

2.315 

75.340 

1.8.0 

.095 

1.932 

6.283 

5.686 

3.142 

2.573 

.569 

1.90912.110 

2.010 

55.965 

21/4 

2.0oO 

.095 

2.186 

7.0o9 

6.472 

3.976 

3.333 

.643 

1  .697  1  .854 

1.775 

43.205 

21/2 

2.282 

.109 

2.783 

7.854 

7.109 

4.909 

4.090 

.819 

1.527 

1.673 

1.600 

35.208 

23/4 

2.532 

.109 

3.074 

8.639 

7.955 

5.940 

5.036 

.904 

1.388 

1.508 

1.448 

28.599 

3 

2.782 

.109 

3.365 

9.425 

8.740 

7.0b9 

6.079 

.990 

1.273 

1.373 

1.323 

23.690 

31/4 

3.010 

.120 

4.011 

10.210 

9.456 

8.296 

7.116 

1.180 

1.175 

1.269 

1.222 

20.237 

31/2 

3.260 

.120 

4.331 

10.996 

10.242 

9.621 

8.347 

1.274 

1.091 

1.171 

1.131 

17.252 

33/4 

3.510 

.120 

4.652 

1  1  .781 

1  1  .027 

1  1  ,045 

9.677 

1.368 

1.018 

1.088 

1.053 

14.882 

4 

3.732 

.134 

5.532 

12.5o6 

1  1  .724 

12.5ob 

10.939 

1.627 

0.954 

.023 

0.989 

13.164 

4l/2 

4.232 

.134 

6.2t8 

14.137 

13.295 

15.904 

14.0b6 

1.838 

.848 

0.902 

.875 

10.237 

5 

4.704 

.148 

7.6o9 

15.708 

14.776 

19.b35 

17.379 

2.256 

.763 

.812 

.787 

8.286 

6 

5.670 

.165 

10.282 

18.850 

17.813 

28.274 

25.249 

3.025 

.636 

.673 

.655 

5.703 

7 

6.670 

.165 

12.044 

21.991 

20.954 

38.485 

34.942 

3.543 

.545 

.572 

.559 

4.12t 

8 

7.670 

165 

13.807 

25.133 

24.096 

50.265 

46.204 

4.061 

.477 

.498 

.487 

3.117 

9 

8.640 

.180 

16.955 

28.274 

27.143 

63.617 

58.629 

4.988 

.424 

.442 

.433 

2.456 

10 

9.594 

.203 

21  .240 

31.416 

30.140 

78.540 

72.292 

6.248 

.381 

.398 

.390 

1.992 

11 

10.560 

.220 

-25.329  34.5^8 

33.175 

95.033 

87.582 

7.451 

.347 

.361 

.354 

1.644 

12 

1  1  .542 

.229 

28.788  37.699 

36.2oO 

113.097 

104.629 

8.468 

.318 

.330 

.324 

1.376 

13 

12.524 

.238 

32.439  40.841 

39.3*5  132.732 

123.190 

9.542 

.293 

.304 

.299 

1.169 

14 

13.504 

.248 

36.424 

43.982 

42.424 

153.938 

143.224 

10.714 

.272 

.282 

.277 

1.005 

15 
16 

14.482 
15.460 

.25940.775 
.270145.359 

47.124 
50.265 

45.497 
48.509 

176.715 
201.062 

164.721 
187.719 

1  1  .994 
13.343 

.254 
.238 

.263 
.247 

.259 
.242 

0.874 
.767 

LAP-WELDED  STEEL  PIPE. 


205 


Weights  and  Bursting  Strength  of  Lap-Welded  Steel  Pipe. 

(American  Spiral  Pipe  Works,  Chicago,  1911.) 

20-Pt.  Lengths,  Plain  Ends  without  Connections.  Thicknesses  in 
U.  S.  Standard  Gage  or  Inches.  Bursting  Strength  in  Lb.  per  Sq.  Jn. 
Internal  Pressure. 


Inside  Dia., 
Ins. 

Thickness, 
Ins. 

d 

i< 

r 

Bursting 
Strength. 

Inside  Dia., 
Ins. 

Thickness, 
Ins. 

3 
s*f 
ft§ 
.pfc 
^ 

Bursting 
Strength. 

Inside  Dia., 
Ins. 

Thickness, 
Ins. 

a 

M 

r 

Bursting 
Strength. 

12 

10G 

19.3 

1172 

28 

3/4 

244 

2678 

42 

1/4 

119 

595 

" 

3/16 

25.8 

1562 

" 

329 

3570 

" 

1/2 

239 

1190 

11 

1/4 

34.6 

2083 

" 

H/4 

416 

4462 

" 

3/4 

362 

1784 

14 

10G 

22.4 

1005 

30 

3/16 

64 

625 

*• 

1 

486 

2380 

" 

V4 

40.2 

1785 

" 

1/4 

85 

833 

" 

1  1/4 

612 

2976 

" 

3/8 

61.0 

2678 

" 

1/2 

172 

1666 

44 

1/4 

124 

568 

11 

1/2 

82.0 

3568 

" 

3/4 

261 

2500 

" 

1/2 

250 

1136 

16 

10G 

25.6 

879 

" 

352 

3328 

" 

3/4 

378 

1705 

" 

I/I 

45.8 

1562 

" 

H/4 

444 

4160 

" 

1 

508 

2277 

" 

3/8 

69.4 

2344 

32 

3/16 

68 

586 

" 

U/4 

640 

2840 

" 

1/2 

93.5 

3124 

" 

1/4 

91 

781 

48 

V4 

135 

520 

M 

5/8 

118.0 

3904 

" 

V2 

183 

1562 

" 

1/2 

273 

1040 

18 

10G 

28.7 

781 

" 

3/4 

278 

2344 

" 

3/4 

412 

1562 

M 

1/4 

51.4 

1388 

" 

1 

374 

3125 

" 

553 

2080 

" 

3/8 

77.8 

2082 

" 

U/4 

472 

3906 

" 

U/4 

696 

2604 

" 

1/2 

104.7 

2776 

34 

3/16 

72 

551 

50 

1/4 

141 

500 

" 

5/8 

132.0 

3472 

" 

1/4 

96 

735 

" 

1/2 

284 

1000 

20 

10G 

31.9 

703 

*• 

1/2 

194 

1470 

" 

3/4 

429 

1500 

M 

1/4 

57.0 

1250 

" 

3/4 

294 

2206 

« 

1 

576 

2000 

" 

1/2 

116.2 

2500 

" 

1 

396 

2942 

" 

11/4 

724 

2500 

•* 

3/4 

177.0 

3736 

" 

U/4 

500 

3678 

54 

1/4 

152 

463 

22 

10G 

35.0 

639 

36 

3/16 

76 

520 

1/2 

306 

926 

" 

1/4 

62.6 

1136 

" 

1/4 

102 

694 

« 

3/4 

462 

1390 

" 

1/2 

127.0 

2272 

" 

1/2 

206 

1388 

" 

1 

620 

1852 

" 

3/4 

194.0 

3410 

« 

3/4 

311 

2080 

" 

U/4 

780 

2315 

" 

1 

262.0 

4555 

'• 

419 

2776 

60 

V4 

169 

416 

24 

10G 

38.0 

586 

« 

U/4 

528 

3472 

1/2 

340 

832 

** 

V4 

68.0 

1041 

38 

s/rt 

80 

493 

'« 

3/4 

513 

1250 

" 

1/2 

138.0 

2082 

« 

1/4 

107 

658 

« 

688 

1664 

" 

3/4 

210.0 

3124 

" 

1/2 

217 

1316 

" 

U/4 

864 

2080 

M 

1 

284.0 

4160 

*< 

3/4 

328 

1972 

66 

1/4 

186 

379 

26 

3/16 

55.0 

721 

" 

441 

2632 

" 

!/2 

374 

758 

1/4 

74.0 

961 

«« 

U/4 

556 

3288 

" 

3/4 

563 

1132 

'* 

1/2 

150.0 

1922 

40 

3/16 

84 

467 

" 

1 

755 

1516 

" 

3/4 

227.0 

2885 

" 

1/4 

113 

625 

'« 

U/4 

948 

1892 

" 

307.0 

3847 

" 

1/2 

228 

1250 

72 

V4 

203 

347 

" 

H/4 

388.0 

4809 

«' 

3/4 

345 

1868 

1/2 

407 

694 

28 

3/16 

60.0 

669 

" 

1 

464 

2500 

*« 

3/4 

614 

1040 

" 

V4 

80.0 

892 

" 

U/4 

584 

3124 

«< 

822 

1388 

" 

1/2 

161  .0 

1784 

42 

3/16 

89 

446 

" 

U/4 

1032 

1736 

For  dimensions  of  extra  heavy  rolled  steel  flanges  for  above  pipe, 
see  table  page  211. 

Square  Pipe,  external  size,  7/g,  1,  H/4,  li/2,  Hl/ie,  2,  21/2,  3  in. 

Rectangular  Pipe,  external  size,  1 1/4  X  1,  11/2X1 V4,  2X1 1/4, 
2X1 1/2,  21/2X1 1/2,  3X2. 

Two  or  more  thicknesses  of  each  size. 

Pipe  Specialties. — Hand  railings  and  their  fittings;  ladders  with  flat 
or  round  pipe  bars  and  runners;  seamless  cylinders,  with  flat,  domed, 
disked,  or  necked  ends;  special  shapes  for  automobiles,  to  replace  drop 
forgings ;  tapered  tubes,  and  other  specialties  are  illustrated  in  National 
Tube  Co.'s  Book  of  Standards. 


206 


MATERIALS. 


Special  Sizes  of  Lap-welded  Pipe — Boston  Casing.   (National  Tube  Co.) 


£    N 

§1 

•«  8 

ss 

tfiS 

5.  a 

li 

68* 

!'! 

M      • 

!§  8 

as 

ll 

£  1 

la 

Is 

ga 

Is 

SQ 

e" 

la 

&* 

E-i  C 

IJ 

*Q 
w 

E-<  fl 

2 

21/4 

0.100 

4l/2 

43/4 

0.145 

55/8 

6 

0.224 

81/4 

85/s 

0.217 

21/4 

21/2 

.108 

41/2 

43/4 

.193 

55/8 

6 

.275 

81/4 

85/8 

.264 

21/2 

23/4 

.113 

43/4 

5 

.152 

61/4 

65/8 

.169 

85/8 

9 

.196 

23/4 

3 

.116 

5 

51/4, 

.153 

61/4 

65/s 

.185 

95/8 

10 

.209 

3 

31/4 

.120 

5 

51/4 

.182 

65/8 

7 

.174 

105/8 

11 

.224 

31/4 

31/2 

.125 

5 

51/4 

.182 

65/8 

7 

.231 

115/8 

12 

.243 

31/2 

33/4 

.129 

5 

51/4 

.241 

7V4 

75/8 

.181 

121/2 

13 

.259 

33/4 

4 

.134 

5 

51/4 

.301 

75/8 

8 

.186 

131/2 

14 

.276 

4 

41/4 

.138 

53/18 

5l/2 

.154 

75/8 

8 

.236 

141/2 

15 

.291 

41/4 

4l/2 

.142 

55/8 

6 

.164 

81/4 

85/8 

.188 

15l/2 

16 

.302 

41/4 

4l/2 

.205 

55/8 

6 

.190 

Other  sizes  of  lap- welded  pipe:  Inserted  Joint  Casing,  external 
diameters  same  as  Boston  Casing,  with  the  least  thickness.  The  5  5/g 
casing  is  made  0.164  and  0.190  in.  thick.  California  Diamond  X  Casing, 
sizes  5  5/8  to  15  1/2,  all  heavier  than  Boston.  Oil  Well  Tubing,  11/4  to  4  in. ; 
Bedstead  Tubing,  3/8  to  3  in.;  Flush  Joint  Tubing,  3  to  18  in.;  Allison 
Vanishing  Thread  Tubing,  2  to  8  in.,  ends  upset,  11/4  to  8  in.,  ends  not 
upset;  Special  Rotary  Pipe,  2  1/2  to  6  in.;  South  Penn  Casing,  53/i6  to 
12 1/2  in. ;  Reamed  and  Drifted  Pipe,  2  to  6  in. ;  Air-line  Pipe,  1 1/2  to  6  in. ; 
Drill  Pipe,  4  to  6  in. ;  Dry-kiln  Pipe,  1  and  1 1/4  in. ;  Tuyere  Pipe,  1  and 
H/4  in. 

TUBULAR  ELECTRIC  LINE  POLES. 

For  railway  work  the  poles  most  used  are  30  ft.  long,  and  are  com- 
posed of  7-in.,  6-in.,  and  5-in.  pipe.  Anchor  poles  are  usually  8-in., 
7-in.,  and  6-in.,  but  often  they  are  made  of  larger  pipe.  Full  directions 
for  designing  such  poles  are  given  in  the  National  Tube  Co.'s  Book  of 
Standards,  which  contains  38  pages  of  tables  of  dimensions,  load,  de- 
flection, etc.,  of  poles  of  different  sizes  and  weights. 

PROTECTIVE  COATINGS  FOR  PIPE. 

(1)  Galvanizing — The  pipe  cleaned  from  scale  and  rust  by  pickling 
in  warm  dilute  sulphuric  acid,  washed,  immersed  in  an  alkaline  bath, 
dried  and  immersed  in  molten  zinc.  (2)  Bituminous  Coating — The 
cleaned,  dried  and  warmed  pipe  is  dipped  in  a  bath  of  refined  coal  tar 
pitch,  free  from  water  and  the  lighter  oils,  at  a  temperature  not  below 
212°,  and  then  baked.  (3)  "National  Coating." — The  bituminous 
coated  pipe,  after  baking  is  wrapped  with  a  strip  of  fabric  saturated 
with  the  hot  compound,  the  edges  of  the  fabric  overlapping. 

VALVES  AND  FITTINGS. 

(From  Information  Furnished  by  National  Tube  Co.,  1915.) 

Wrought  pipe  is  usually  connected  in  one  of  three  ways,  screwed, 
flanged  or  leaded  joints. 

Screwed. — Pipe  in  sizes  from  i/g  m.  to  15  in.  inclusive  is  regularly 
threaded  on  the  ends,  and  is  connected  by  means  of  threaded  couplings. 

Flanged. — Pipe  in  sizes  11/4  inches  and  larger  is  frequently  connected 
by  drilled  flanges  bolted  together,  the  joint  being  made  by  a  gasket 
between  the  flange  faces. 

Flanges  are  attached  to  the  pipe  in  a  variety  of  ways.  The  most 
common  method  for  sizes  of. pipe  from  U/4  in.  to  15  in.  inclusive 
is  by  screwing  them  on  the  pipe.  Many  prefer  peened  flanges  for 
pipe  larger  than  6  in.  The  peened  flange  is  shrunk  on  the  end  of 
the  pipe,  and  the  latter  is  then  peened  over  or  expanded  into  a  recess 
in  the  flange  face.  Steel  flanges  are  also  welded  to  pipe  and  loose 
flanges  are  used  by  flanging  over  the  pipe  ends.  When  no  method 
of  attaching  is  stated,  screwed  flanges  are  always  furnished. 


VALVES  AND   FITTINGS.  207 

Working  Pressures. — All  valves  and  fittings  are  classified,  as  a  rule, 
under  five  general  headings,  representing  the  working  pressures  for 
which  they  are  suitable,  as  follows:  Low  Pressure,  up  to  25  pounds 
per  square  inch.  Standard,  up  to  125  pounds  per  square  inch.  Medium 
Pressure,  from  125  pounds  to  175  pounds  per  square  inch.  Extra 
Heavy,  from  175  pounds  to  250  pounds  per  square  inch.  Hydraulic, 
for  high  pressure  water  up  to  800  pounds  per  square  inch. 

The  following  table  gives  the  names  of  different  valves  and  fittings, 
the  material  of  which  they  are  made,  and  the  regular  sizes  manu- 
factured for  the  different  pressures  (L,  low;  S,  standard;  M,  medium; 
E,  extra  heavy ;  H ,  hydraulic) : 

SCREWED  FITTINGS. 

Malleable  Iron S,  E,  H,  sizes  1/8  to    8  in. 

Cast  Iron S,  E,  1/4  to  12  in. 

FLANGED  FITTINGS. 

Cast  Iron L,  S,  E,  H,  sizes  2  in.  and  larger. 

GATE  VALVES. 

Brass L  S  M  E        If  up  to  3  in. 

Iron  Body,  sizes. .    12  to  48     2  to  30  2  to  18    1 1/4  to  24    H/2  to  12  in. 

GLOBE  AND  ANGLE  VALVES. 

Brass S,  i/s  to  4;  M,  1/4  to  3;  E,  1/2  to  3;  H,  1/2  to  2 

Iron  Body S,  2  to  12;  E,  2  to  12 

CHECK  VALVES. 

Brass S,  M,  E,  H,  sizes  l/s  to    3  in. 

Iron  Body L,  S,  M ,  E,  H,     '      2    to  12  in. 

COCKS,  STEAM  AND  GAS. 

Brass sizes  1/4  to  3  in. 

Iron  Body *      1/2  to  3  in. 

Nipples. — Nipples  are  made  in  all  sizes  from  i/g  in.  to  12  in.  in- 
clusive, in  all  lengths,  either  black  or  galvanized,  and  regular  right- 
hand  or  right-  and  left-hand  threads.  (For  table  of  nipples  see  National 
Tube  Co.'s  Book  of  Standards.)  Long  screws  or  tank  nipples  are  made 
of  extra  heavy  pipe  because  there  is  less  danger  of  crushing  or  splitting 
them  when  screwing  up. 

Screwed  Fittings — Malleable  Iron. — Standard  Malleable  Iron  Fittings 
are  made  both  plain  and  beaded.  The  former  are  generally  used  for 
low  pressure  gas  and  water,  as  in  house  plumbing  and  railing  work.  The 
beaded  is  the  standard  steam,  air,  gas,  or  oil  fitting.  Beaded  fittings, 
in  sizes  4  in.  and  smaller,  are  made  in  nearly  every  useful  combination  of 
openings.  Sizes  larger  than  4  in.  are  not  usually  made  reducing  except 
by  means  of  bushing.  Extra  heavy  and  hydraulic  malleable  iron 
fittings  are  flat  bead,  or  banded. 

Screwed  Fittings — Cast  Iron. — It  is  not  considered  good  practice  to 
use  screwed  cast-iron  fittings  of  any  kind  in  sizes  larger  than  6  in. 

Flanged  Fittings. — The  flanges  of  the  low  pressure  and  standard  are 
the  same  with  the  exception  of  the  flange  thickness,  which  is  less  on  the 
low  pressure.  These  flanges  are  known  as  the  American  Standard. 
(See  pp.  209,  210.) 

There  is  no  recognized  standard  for  flanges  in  hydraulic  work. 

Unions. — Unions  are  usually  classified  under  two  headings,  Nut  unions 
and  Flange  unions.  Nut  unions  are  commonly  used  in  sizes  2  in.  and 
smaller,  and  flange  unions  in  sizes  larger  than  2  in.  However,  many 
manufacturers  make  nut  unions  as  large  as  4  in.  and  flange  unions 
smaller  than  2  in. 

Nut  unions  are  made  in  malleable  iron,  brass,  and  malleable  iron, 
and  ail  brass.  The  all  malleable  iron  union  (lip  union)  is  the  standard 
malleable  iron  union  of  the  trade  and  requires  a  gasket.  The  brass 
and  malleable  iron  union  is  a  better  union,  because  no  gasket  is  re- 
quired and  there  is  no  possibility  of  the  parts  rusting  together.  The 
pipe  end  of  this  union  which  carries  an  external  thread,  called  the 


208  MATERIALS. 

thread  end,  upon  which  the  ntit  or  ring  screws,  is  made  of  brass,  and  the 
other  pipe  end  (called  the  bottom)  and  nut  ring  are  made  of  malleable 
iron.  The  seat  formed  by  the  brass  and  iron  pipe  ends,  when  brought 
together,  is  truly  spherical  and  the  harder  iron  is  sure  to  make  a  perfect 
joint  with  the  softer  brass. 

All-brass  unions  are  made  with  a  spherical  or  conical  seat,  no  gaskets 
being  required.  The  finished  all-brass  union  is  often  used  where  showy 
work  is  desired,  such  as  oil  piping  for  engines,  etc. 

Flange  unions  are  made  of  malleable  iron,  malleable  iron  and  brass, 
cast  iron,  and  cast  iron  and  brass. 

The  type  of  flange  union  recommended  for  standard  work  is  made 
with  a  brass  to  iron  non-corrosive  ball  joint  seat  which  requires  no 
gasket  to  make  a  tight  joint  even  when  the  pipe  alignment  is  imperfect. 
The  flange  is  loose  on  the  collar,  so  that  the  bolts  match  the  holes  in 
any  position. 

Valves  and  Cocks. — The  most  common  means  for  regulating  the  flow 
of  fluids  in  pipes  is  by  means  of  valves  and  cocks,  valves  being  pre- 
ferred because  of  the  easier  operation  and  greater  reliability.  The 
common  types  of  valves  are  straightway  or  gate,  globe,  and  angle.  A 
globe  valve  offers  more  resistance  to  the  flow  of  any  fluid  than  the 
straightway  valve. 

Globe  and  Angle  Valves. — Many  manufacturers  make  a  globe  and 
angle  valve  known  as  light  standard  or  competition  valve,  but  it  is 
not  recommended  for  any  work  except  the  lowest  pressures,  or  where 
the  valve  will  not  be  often  opened  or  closed. 

Cocks. — Among  the  modern  types  of  cocks  is  one  made  with  iron 
body  and  brass  plug.  This  cock  has  an  inverted  plug  with  a  spring 
at  the  bottom  constantly  pressing  the  plug  against  the  seat,  which 
reseats  the  plug  if  it  should  stick.  These  cocks  are  tested  to  250  Ib. 
cold-water  pressure,  and  125  Ib.  compressed-air  pressure  under  water, 
and  are  recommended  for  125  Ib.  working  pressure. 

Blast  Furnace  Fittings. — Tuyere  cocks  and  tuyere  unions  used  in 
blast  furnace  piping  are  always  made  of  brass  on  account  of  ease  in 
disconnecting,  greater  reliability  of  metal  and  resistance  to  corrosion 
from  the  impurities  in  the  water,  such  as  sulphuric  acid. 

STANDARD   PIPE   FLANGES   (CAST  IRON). 

The  following  tables  showing  dimensions  of  standard  pipe  flanges 
were  adopted  by  the  American  Society  of  Mechanical  Engineers,  the 
Master  Steam  and  Hot  Water  Fitters'  Association,  and  a  committee 
representing  the  manufacturers  of  pipe  fittings.  They  represent  a 
compromise  between  the  standards  adopted  by  the  American  Society  of 
Mechanical  Engineers  and  the  Master  Steam  and  Hot  Water  Fitters' 
Association  hi  1912,  known  as  the  1912  U.  S.  Standard,  and  the  stand- 
ards adopted  by  a  conference  of  manufacturers  in  July,  1912,  known 
as  the  Manufacturers'  standard.  The  new  standards,  given  in  the 
tables,  are  called  the  American  Standard,  and  became  effective  Jan.  1, 
1914.  The  table  of  flanges  for  extra  heavy  fittings  is  for  working 
pressures  up  to  250  Ib.  per  sq.  in.  The  table  for  ordinary  fittings  is  for 
working  pressures  up  to  125  Ib.  per  sq.  in.  In  the  tables,  the  values  of 

T  X  T) 

stresses  in  pipe  walls  were  calculated   from  the  formula  S  =  - — .—  > 

where  p  =  working  pressure,  Ib.  per  sq.  in.,  t  =  thickness  of  pipe, 
in.,  and  r  =  radius  of  pipe,  in.  The  highest  stress  was  found  to  be 
2000  Ib.  per  sq.  in.  on  the  250-lb.,  46-  and  48-in.  pipe  walls,  giving  a 
factor  of  safety  of  about  10.  The  desirable  thickness  of  pipe  (Col.  2) 

is  calculated  from  the  formula  T  =  PA* 3°  -P  +  0.333/1  -  -^  Jl.2. 

where  p  =  pressure,  Ib.,  per  sq.  in.,  5  =  1800,  and  d  =  diameter 
of  pipe.  The  minimum  thickness  in  even  fractions  of  an  inch  is  given 
in  Col.  3.  The  following  approximate  formulae  were  also  used  for 
ordinary  fittings:  Diam.  of  bolt  circles  =  1.10  d  +  3.  Flange  thick- 
ness (for  pipe  diameters  26  to  100  in.  inclusive)  =  0.0315  d  +  1.25. 
For  extra  heavy  fittings  the  formulae  are:  Bolt  circle  =  1.171d+3.75; 
Flange  thickness  =  0.0546  d  +  1.375  (for  sizes  10  to  48  in.  inclusive). 


American  Standard  Cast  Iron  Pipe  flanges  for  Pressures  Up  to  ™ 

Lb.  per  Sq.  In.      (All  Dimensions  in  Inches.) 


r  r    Pipe 

£« 

Flanges. 

Bolts. 

fc 

Thickness 

E  • 

jj 

i 

o 

Jj 

^ 

£ 

.S 

a£ 

f- 

See  Fig.  75, 
p.  210 

S 

& 

|i 

n  & 

W    OH 

! 

Q 

IS 
H 

if 

0) 

g 

5 

I! 

CU     *M 

g& 

A 

B 

C 

1 

0.43 

7/16 

143 

4 

7/16 

I  1/2 

3 

~4 

7/16 

0.093 

264 

9/16 

2.12 

^9l 

U2T 

1  1/4 

0.44 

7/16 

178 

41/2 

1/2 

15/8 

33/8 

4 

7/16 

0.093 

412 

9/16 

2.38  0.91 

L47 

H/2 

0.45 

7/16 

214 

9/16 

13/4 

37/8 

4 

1/2  0.126; 

438 

5/82.731.00 

1.73 

2 

0.46 

7/16 

286 

6 

5/8 

2 

43/4 

4 

5/81 

0.202 

486 

3/4! 

3.35 

1.21 

2.14 

21/2 

0.48 

7/16 

357 

7 

11/16 

21/4 

51/2 

4 

5/8  0.202 

750 

3/4  3.88 

1.21 

2.67 

'3 

0.50 

7/16 

428 

71/2 

3/4 

21/4 

6 

4 

5/8  i  0.202 

1093 

3/4 

4.23 

1.21 

3.02 

31/2 

0.52 

7/16 

500 

81/2 

13/16 

21/2 

7 

4 

5/8 

0.202 

1488 

3/4 

4.94 

1.21 

3.73 

4 

0.53 

1/2 

500 

9 

15/16 

21/2 

71/2 

8 

5/8  '0.202 

972 

3/4 

2.87 

1.21 

1.56 

41/2 

0.55 

1/2 

562 

91/4 

15/16 

23/8 

73/4 

8 

3/4 

0.302 

823 

7/8^2.96 

1.44 

1.52 

5 

0.56 

1/2 

625 

10 

15/16 

21/2 

81/2 

8 

3/4 

0.302 

1016 

7/8 

3.25 

1.44 

1.81 

6 

0.60 

9/16 

667 

11 

21/2 

91/2 

8 

3/4 

0.302 

1463 

7/8 

3.63 

1.44 

2.19 

7 

0.63 

5/8 

700 

12l/2 

1/16 

23/4 

103/4 

8 

3/4  '0.302 

1991 

7/84.11 

1.44 

2.67 

8 

0.66 

5/8 

800 

131/2 

1/8 

23/4 

113/4 

8 

3/4!  0.302  2600 

7/8  4.50 

1.44 

3.06 

9 

0.70 

H/16 

818 

15 

1/8 

3 

131/4112 

3/4  0.302  2194 

7/8|3.43 

1  .44  1  .99 

10 

0.73 

3/4 

833 

16 

8/18 

3 

141/4112 

7/8 

0.420 

1948 

3.69 

1  .66  2.03 

12 

0.80 

13/16 

923 

19 

1/4 

31/2 

17 

12 

7/8 

0.420 

2805 

4.40 

1.66 

2.74 

14 

0.86 

7/8 

1000 

21 

3/8 

31/2 

183/4 

12 

0.5502915 

1/8 

4.86  1  .88 

2.98 

15 

0.90 

7/8    1072 

221/4 

3/8 

35/8 

20 

16 

1 

0.5502510 

1/83.90  1.88 

2.02 

16 

0.93 

1000 

231/2 

7/16 

33/4 

21  1/4 

16 

1 

0.550 

2856 

1/8 

4.14 

1.88 

2.26 

18 

1.00 

1/16 

1059 

25 

9/16 

31/2 

223/4 

16 

1  l/s  0.694 

2865 

1/44.44 

2.09 

2.35 

20 

1.07 

1/8 

1111 

271/2 

H/16 

33/4 

25 

20 

1  1/8 

0.694 

2829 

1/4 

3.91 

2.09 

1.82 

22 
24 

1.13 
1.20 

3/16 
1/4 

1158 
1200 

13/16 
7/8 

33/4 

271/4 
291/2 

20 
20 

1  1/4  0.893  2660 
l/iO.8933166 

1  3/8  4.2612.31 
1  3/8;4.62l2.31 

1.95 
2.31 

26 

1.27 

5/16 

1238 

341/4 

2 

41/8 

313/4 

24 

1  1/4 

0.893 

3096 

13/8 

4.14 

2.31 

1.83 

28 
30 

1.33 
1.40 

3/8 
7/16 

1273 
1304 

361/2 

383/4 

2    1/16 
2    1/8 

4i/4|  34  ' 
43/8l  36 

28 
28 

1  l  /4  10.  893  1  3078 
13/8  1.057  12985 

1  3/8  3.81 
1/2  4.03 

2.31 
2.53 

1.50 
1.50 

32 

1.47 

1/2 

1333 

413/4J2    1/4 

47/8 

381/2 

28 

1  1/2 

1.294 

2775 

5/8 

4.31 

2.75 

1.56 

34 

1.54 

9/16 

1360 

433/42   5/i6 

47/8 

401/2 

32 

H/2 

.294274 

1  5/8  3.97 

2.75 

1.22 

36 

1.60 

5/8 

1385 

46 

2    3/8 

5 

423/4 

32 

1  1/2 

.294 

3073 

15/8 

4.19 

2.75 

1.44 

38 

1.67 

H/16 

1407 

483/4 

2  3/8 

53/8 

451/4 

32 

1  5/8 

.515 

2924 

1  3/4  4.43 

2.96 

1.47 

40 

1.73 

3/4 

1428 

503/42    1/2 

53/8    471/4 

36 

1  5/8 

.515 

2880 

13/4 

4.11 

2.96 

1.15 

42 

1.82 

13/16 

1448 

53 

2    5/8 

5l/2    49i/2 

36 

1  5/8 

.5153175 

13/4 

4.31 

2.96 

1.35 

44 

1.87 

7/8  11467 

551/4 

2    5/8 

5  5/8    51  3/4 

40 

15/8 

.515 

3136 

1  3/4  4.06 

2.96 

1.10 

46 

1.94 

115/ie  1484 

571/42H/16 

55/8    533/4 

40 

1  5/8 

.515 

3428 

18/4 

4.22 

2.96 

1.26 

48 

2.00 

2 

1500 

591/212  3/4 

53/4 

56 

44 

1  5/8 

.515 

3393 

13/4 

3.98 

2.96 

1.02 

50 
52 

2.07 
2.14 

21/16 
21/8 

1515 
1530 

SI'" 

2   3/4 
2   7/8 

57/8 
6 

581/444 
60  1/2  44 

13/4J   .746)3195 
1  3/4{   .746  3456 

7/8|4.14 
7/84.30 

3.19 
3.19 

0.95 
1.11 

54 

2.20 

23/16  1543   661A3         161/s 

62  3/4  44 

13/4 

.746 

3726 

17/8 

4.45 

3.19 

1.26 

56 

2.27 

21/4 

1555    683/4 

3           163/8 

65 

48 

13/4 

.746 

3674 

1  7/8  4.26 

3.19 

1.07 

58 

2.34 

2  5/16  1567 

71 

3   1/8    61/2 

671/4 

48 

1  3/4 

.746 

394 

17/fi 

4.4013.19 

1.21 

60 

2.41 

27/ie  1538 

73 

3    1/8    61/2 

691/4 

52 

13/4 

.7463892 

1  7/8  4.19 

3.19 

1.00 

62 

2.47 

2  1/2    1550 

753/4 

3   1/4 

67/8 

713/4 

52 

1  7/8  12.  051  3538 

2 

4.34 

3.41 

0.93 

64 

2.54 

2  9/16 

1561 

78 

3    1/4 

7 

74 

52 

7/8 

2.051 

3770 

2 

4.48 

3.41 

1.07 

66 

2.61 

25/8 

1572 

80 

3  3/g 

7 

76 

52 

7/82.051 

4010 

2 

4.60 

3.41 

1.19 

68 

2.68 

2H/16H582 

821/43   3/8 

71/8 

781/4 

56 

7/8 

2.051 

3952 

2 

4.38 

3.41 

0.97 

70 

2.74 

23/4  11591 

84l/213  1/2 

71/4 

801/2 

56 

7/8  !  2.  051 

4188 

2 

4.51 

3.41 

1.10 

72 

2.81 

213/ie  '1600    86  1/2  3   1/2 

71/4 

821/2 

60 

7/82.051 

4136 

2 

4.33 

3.41 

0.92 

74 

2.88 

27/8    1609    881/213  5/8 

71/4 

841/2 

60 

7/8 

2.051 

4368 

2 

4.44 

3.41 

1.03 

76 

2.94 

215/16 

1617    903/43  5/8 

73/8 

861/260 

7/82.051 

4608 

2 

4.54 

3.41 

1.13 

78 

3.01 

3 

1625    93    - 

3  3/4 

71/2 

883/4 

60 

2 

2.302 

432 

21/8 

4.66 

3.63 

1.03 

80 

3.08 

31/16 

1633    951/4 

3  3/4 

75/8 

91 

60 

2 

2.302 

4549 

2l/8!4.78 

3.63 

1.15 

82 

3.15 

31/8 

1640    971/23   7/8 

73/4    931/4 

60 

2 

2.302 

4779 

2  l/s  4.90 

3.63 

1.27 

84 

3.21 

33/16 

1647    993/43  7/8 

77/8    951/2 

64 

2 

2.302 

4702 

21/8 

4.68 

3.63 

1.05 

86 

3.28 

31/4 

1653  102 

4 

8        973/4 

64 

2 

2.302 

4928 

2  l/8,'4.79 

3.63 

1.16 

88 

3.35 

35/16 

1660  1041/4 

4 

81/8  100 

68 

2 

2.302 

4857 

2  l/s 

4.60 

3.63 

0.97 

90 

3.41 

33/8 

1667s  106  1/2  4  1/8 

81/411021/4 

6821/s 

2.648  4416 

21/4 

!4.71 

3.83 

0.88 

92 

3.48 

31/2 

16431083/44  l/s 

83/8  104  1/2  68  21/8^2.648  4615 

2  1/4  4.81 

3.83 

0.98 

94 

3.55 

39/16 

1649  111 

4    1/4 

8.1/2  1061/468  21/8 

2.648 

4817 

21/4 

4.89 

3.83 

1.06 

96 
98 
100 

3.62 
3.68 
3.75 

35/8 
3H/16 
33/4 

1655  1131/4 
1661  1151/s 
1667J1173/4 

41/4    85/81081/216821/43.023440 
4  S/g  Is  3/4  110  3/4|68  2  1/4  3.023  4587 
4  3/8  |87/8;113      |68l2  1/413.023  4776 

2  3/8  4.99  4.06 
23/85.094.06 
23/85.20l4.06 

0.93 
1.03 
1.14 

210 


MATERIALS. 


The  last  three  columns  of  the  table  refer  to  the  sketch  Fig.  75,  and  show 

the  distances  between  bolt  holes,  the  maximum 

space  occupied   by  the  nuts  and  the  minimum 

t-*-B->j       space   between   adjacent   nuts,   all  measured  on 

/-f-\       '/i~\!       tne  cnord-     Bolt  holes  are  to  straddle  the  center 

/.  ;   \ — !(--}— V-      une'  ancl  are  to  De  Vs  in.  larger  in  diameter  than 

\     /^CJ\     /        the  bolts.     Standard  weight  fittings  and  flanges 

j~     ^  are  to  be  plain  faced,  but  extra  heavy  fittings  and 

flanges  are  to  have  a  raised  surface  i/ie  in.  high 

(On  Chord)  inside  of  bolt  holes  for  gaskets.    Square  head  bolts 

with  hexagonal  nuts  are  recommended,  but  for 

Fig.  75.  bolts  is/g  in.  diameter  and  larger,  studs  with  a  nut 

on  each  end  may  be  substituted.     Flanges  are  to 

be  spot  bored  for  nuts  for  sizes  32  in.  to  100  in.  inclusive.     For  super- 
heated steam,  steel  flanges,  fittings  and  valves  are  recommended. 

American  Standard  Extra  Heavy  Cast  Iron  Pipe  Flanges 
For  Pressures  up  to  250  Lb.  per  Sq.  In.     (All  Dimensions  in  Inches.) 


Pipe             |d 

Flanges. 

Bolts. 

See  Fig.  75, 
p.  210. 

g 

Thickness.^ 

i 

"8 

1 

. 

fe 

d 

•  d 

0) 

o 

8 

ij 

•sl 

•I 

w  P< 

i 

Thickn. 

8 

0  g 

|  Numbe 

IA 

>  a1 
43  co 

*»  °* 

4J>  rt 

55 

PQ 

A 

B 

C 

~y 

0.45 

1/2 

250 

41/2 

H/16 

13/4    31/4 

4     1/2 

0.126 

389 

5/8 

2.29 

.00 

1.29 

H/4 

0.47 

1/2 

312 

5 

3/4 

17/8 

33/4    4!     1/2 

0.126 

609 

5/8 

2.65 

.00 

1.65 

,  U/2 

0.49 

1/2 

375    6 

13/18 

21/4 

41/2    4    5/80.202 

547 

3/43.17 

.21 

1.96 

0.51 

1/2 

500 

61/2 

7/8 

21/4 

5 

4     5/8 

0.202 

972 

3/4 

3.53 

.21 

2.32 

21/2 

0.53 

9/16 

555 

71/2 

|      ' 

21/2 

57/g 

4 

3/4,0.302 

1016 

7/8 

4.15 

.442.71 

3 

0.56 

9/16 

667 

81/4 

1  1/8 

25/8 

65/8!  8 

3/4 

0.302 

731 

7/8 

2.53 

.44 

.09 

31/2 

0.59 

9/16 

778 

9 

13/16 

23/4|   71/4|   8 

3/40.302 

995     7/8;2.77 

.44 

.33 

0.61 

5/8 

800 

10 

1  1/4 

3 

77/8 

0 

3/4 

0.302 

1300!    7/8 

3.01 

.44 

.57 

41/2 

0.64 

5/8 

900 

101/2 

1  5/16 

3 

81/2 

8 

3/t  0.302 

1646     7/8  '3.  25 

.44 

.81 

5 

0.67 

909 

11 

13/8 

3 

91/4 

8 

3/4!0.302|2032 

7/8  3.53 

.44  2.09 

6 

0.72 

3/4 

1000 

121/2 

1  7/16 

31/4 

105/8 

12 

3/4  0.302 

1950     7/82.75 

1.44 

.31 

7 

0.78 

13/16 

1077 

14 

31/2J  11  7/8 

12 

7/80.420 

1909 

3.07 

1.66 

.41 

8 

0.83 

13/16 

1230  15 

15/8 

31/213 

12 

7/8  0.420  2493 

3.36 

1.66 

.70 

9 

0.89 

7/8 

1285 

161/4 

1  3/4 

35/8:i4 

12 

1 

0.550 

2410     l/s 

3.62 

1.88 

.74 

10 

0.94 

15/16 

1333 

171/2 

17/8 

33/4151/4 

16 

1 

0.5502231     1/8,2.97 

1.88 

.09 

12 

1.05 

| 

1500 

201/2 

2 

41/4 

173/4 

16 

11/80.6942546     1/4  3.46 

2.09 

.37 

14 
15 
16 

1.16 
1.21 
1.27 

U/8 
13/16 

HA 

155523 
1579  24  1/2 

1600:25  1/2 

21/8 
23/16 
21/4 

41/2!20l/420  1  i/sO.6942773     1/4  3.17  2.09 
43/4  21  1/2  20  1  1  A  1  0.  893  1  2473!    3/8  3.36  2.31 
43/4J22  1/2120  1  1/4  10.  893  2814    3/8|3.52  2.31 

.08 
.05 
.21 

18 

1.37 

13/8 

1636 

28 

23/8 

5 

243/4 

24 

1  1/4  0.893  j  2968     3/8|3.232.31 

0.92 

20 
22 

1.48H/2     1666301/2 
1.5919/16    176033 

21/2 
25/8 

5  1/4  27 

5  1/4  29  1/4 

24 
24 

13/8 
1  1/2 

.057  3096 
.295  3058 

1/2  3.52  2.53  0.99 
5/8  3.81  2.75    .06 

24 

1  .70  1  5/8 

1846 

36 

23/4 

5  3/4  32 

24 

1  5/8 

.515 

3110il  3/4 

4.18296 

.22 

26 

1.81  1  13/ie 

1793381/4 

2  13/16 

61/8341/2 

28 

15/8 

.5153126  1  3/4  3.86!  2.96  0.9  J 

28 

1.91 

17/8 

1866 

403/4 

215/1663/8 

37 

28 

15/8 

.515!3629j1  3/44.142.96 

.18 

30 

2.02 

2 

1875 

43 

3 

61/2 

391/4 

28 

1  3/4 

1.7463615  1  7/8!4.38  3.19 

.19 

32 

2.1321/s.     1882451/4 

31/8 

65/841  1/228 

1  7/82.051 

3501  2 

4.64,341 

.26 

34 
36 

2.2421/4    1  1889  47  1/2 
2.35123/s     189450 

31/4 
33/8 

63/4 

43i/2 
46 

28 
32 

1  7/8  2.051 
1  7/82.051 

39522 

38772 

4.873.41 
4.503.41 

.46 
.09 

38 

2.4627/16    1948521/4 

37/ie  71/848 

32  1  7/8  2.051 

43202 

4.703.41 

.29 

40 
42 
44 
46 
48 

2.562  9/161953541/2 
2.67  2  n/16  1953  57 
2.78:213/161955591/1 
2.8912  7/8   200061  1/2 
3.003         :  2000  65 

3    9/i67l/4 
3  11/16  7  1/2 
33/4     .75/8 
37/8      73/4 
4          l81/2 

501/4  36  1  7/8!2.051  4255  2       4.38  3.41  0.97 
523/4J36  1  7/8  2.051  4691  2        4.59  3.41    .18 
55       1362       2.302458721/84.793.63    .16 
57  1/4  40  2       ;2.302  4512  2  1/8  4.49  3.63  0.86 
603/4402       12.302  4913  2  i/s  4.76  3.63  1.13 

*  Thickness  of  flange  given  in  table  includes  raised  face. 


FORGED  AND  ROLLED  STEEL  FLANGES. 


211 


Forged  Steel  Flangeslfor  Riveted  Pipe. 

Riveted  Pipe  Manufacturers'  Standard.* 


ll 

II 

£72 

0) 
T3   g 

3  i 
£5 

Thickness 
of  Flange.* 

•83 

|3 

•3<2 

IS 

Diam.  of 
Bolt  Circle. 

-2 

g£ 
11 

£W 

Outside 
Diam. 

Thickness 
of  Flange.* 

o» 

& 

0-2 

IS 

Diam.  of 
Bolt  Circle. 

4 
5 
6 
7 
8 
9 
10 
j  ] 

6 

8 
9 
10 
11 
13 
14 
J5 

5/16  .... 
5/16  9/16 
5/16  9/16 
3/8  9/16 
3/8  9/i6 
3/8  5/8 
3/8  5/8 
3/8  11/16 
7/16 

4 
8 
8 
8 
8 
8 
8 
8 
12 

V/18 

7/16 
7/16 
1/2 
1/2 
1/2 
1/2 
1/2 
1/2 

43/4 
5  15/16 
6  15/16 

7T,, 

10 

M   1/4 
121/4 
13  3/8 

16 
18 
20 
22 
24 
26 
28 
30 
32 

211/4 

%,\ 
%'< 

32 
34 
36 
38 

V8        3/4 
V8        3/4 
5/8        V8 
H/16      7/8 
11/16      7/8 

1 

1 

12 
16 
16 
16 
16 
24 
28 
28 
28 

V2 
5/8 
5/8 
5/8 

V8 
3/4 

3/4 
3/4 
3/4 

191/4 
2H/4 

231/8 
26 
273/4 
293/4 
313/4 
333/4 
353/4 

12 
13 
14 
15 

16 
17 
18 
19 

7/16  3/4 
7/16  .... 
7/16  3/4 
9/16  3/4 

12 
12 
12 
12 

1/2 
1/2 
1/2 
1/2 

141/4 

15  1/4 

161/4 

177/16 

34 
36 
40 
42 

40 
42 
46 
48 

H/8 

H/8 
M/8 

28 
32 
32 
36 

3/4 
3/4 
3/4 

3/4 

373/4 
393/4 
433/4 
453/4 

*  Flanges  for  riveted  pipe  are  also  made  with  the  outside  diameter  and 
the  drilling  dimensions  the  same  as  those  of  the  A.  S.  M.  E.  standard 
(page  209) ,  and  with  the  thickness  as  given  in  the  second  column  of  fig- 
ures under  "Thickness  of  Flange"  in  the  above  table. 

Curved  Forged  Steel  Flanges  are  also  made  for  boilers  and  tanks. 
See  catalogue  of  American  Spiral  Pipe  Works,  Chicago. 

Forged  and  Rolled  Steel  Flanges. 

Dimensions  in  Inches.     (American  Spiral  Pipe  Works,  1913.) 


Standard  Companion  Flanges. 

Standard  Shrink  Flanges. 

"3          DD 

•g 

•8 

-.                 . 

•3 

•8 

1      .| 

T3  PJ 

•a  a? 

1 

^3    • 

S3        8 

-8  a 

a 

i     , 

fd   • 

.  . 

|l 

OQ 

|i 

p 

|I 

n 

5w 

III 

l| 

iP 

"a^ 

|1 

A 

B 

C 

D 

E 

A 

B 

C 

D 

E 

2 

6 

21/8 

5/8 

1 

31/8 

4 

9 

43/8 

15/16 

23/i6 

53/4 

21/2 

7 

21/2 

H/16 

1  Vl6 

35/8 

41/2 

91/4 

47/8 

15/16 

21/4 

61/8 

^ 

71/3 

31/8 

3/4 

1  1/8 

45/16 

5 

10 

57/16 

15/16 

25/i6 

67/8 

31/2 

81/2 

35/8 

13/16 

13/16 

47/8 

6 

It 

61/2 

27/ie 

77/8 

4 

9 

41/8 

15/16 

13/16 

53/8 

7 

121/2 

71/2 

1/16 

21/2 

9 

41/2 

91/4 

45/8 

15/16 

5  13/16 

8 

131/2 

81/2 

1/8 

25/8 

10 

5 

10 

51/8 

15/16 

1  5/16 

67/ie 

9 

15 

91/2 

1/8 

23/4 

11  1/8 

6 

11 

I  7/16 

7  9/16 

10 

16 

Id  5/8 

3/16 

3 

121/4 

7 

121/2 

7  3/1J 

1/16 

H/2 

85/8 

12 

19 

125/8 

1/4 

33/8 

141/2 

8 

131/2 

1/8 

15/8 

9  H/16 

14 

21 

137/s 

3/8 

33/8 

157/s 

9 

15 

9  3/i6 

1/8 

13/4 

105/s 

15 

221/4 

147/8 

3/8 

31/2 

167/8 

10 

16 

105/18 

3/16 

1  7/8 

1  1  15/16 

16 

231/2 

157/8 

7/16 

35/8 

18 

12 

19 

125/i6 

1/4 

21/16 

141/8 

18 

25 

177/s 

9/16 

37/8 

201/8 

14 

21 

131/2 

13/8 

157/16 

20 

271/2 

197/s  J 

11/16 

41/8 

22  1/4 

212 


MATERIALS, 


Forged  and  Rolled  Steel  Flanges.— Continued. 
Extra  Heavy  Companion  Flanges. 


Is 
iff 

233 

Outside 
Diam. 

Js 

Thick- 
ness. 

*o  . 

J3-^5 

•s  3 

1* 

*8  . 

c-a 

I* 

Nominal 
Size,  Ins. 

Outside 
Diam. 

.1 

IQ 

Thick- 
ness. 

o 
£•§ 

1* 

i 

# 

A 

B 

C 

D 

E 

A 

B 

C 

D 

E 

91/8 

101/8 

113/16 
129/ia 
145/8 
1513/w 
17  3/i6 
181/4 

j* 

31/2 
4l/2 
6 

61/2 

71/2 
81/4 

9 
10 

,o./, 

121/2 

21/8 
21/2 
31/8 
35/8 
41/8 
45/8 
51/8 
63/16 

7/8 

1/8 
1/8 
1/4 
1/4 
H/4 

3/8 
7/16 
9/16 
5/8 
3/4 
13/16 

2?/8 

33/8 
41/16 
4H/16 
55/16 
5  13/16 
61/4 
6  13/16 
77/8 

7 

8 
9 
10 
12 
14 
15 
16 

14 
15 
16 

171/2 

20 

221/2 

£1/2 

73/16 
83/i6 
93/16 
105/ie 
I25/16 

131/2 
141/2 
151/2 

1  5/16 
13/8 
17/16 
H/2 
1V8 
13/4 
1  13/16 
17/8 

21/16 
23/16 
21/4 
23/8 
29/16 
2  H/16 
2  13/16 
31/16 

Extra  Heavy  High  Hub  Flanges. 


Size. 

A 

B 

C 

D 

E 

Size. 

A 

B 

C 

D 

E 

4 

10 

43/8 

I  1/8 

31/8 

53/4 

18 

27 

177/8 

2 

5 

203/4 

"  4  1/2 

101/2 

47/8 

H/4 

31/4 

61/4 

20 

291/2 

197/g 

21/4 

5l/2 

22i/2 

5 

11 

57/16 

H/4 

31/4 

7 

22 

3H/2 

21/4 

51/2 

243/4 

6 

12l/2 

61/2 

H/4 

31/4 

7  15/16 

24 

34 

27/16 

61/4 

27 

7 

14 

71/2 

15/16 

33/8 

.91/8 

30 

40 

27/W    61/4 

33 

8 

15 

81/2 

13/8 

3l/2 

105/i6 

36 

46 

2  7/16 

61/4 

39 

9 

16 

91/2 

17/16 

35/8 

113/8 

42 

52 

27/16    61/4 

45 

10 

17V2 

105/g 

H/2 

33/4 

125/s 

48 

581/4 

27/16 

61/2 

5H/4 

11 

183/4 

115/8 

19/16 

37/8 

135/8 

54 

641/2 

27/16 

61/2 

571/4 

12 

20 

125/8 

15/8 

4 

143/4 

60 

703/s 

27/16 

.61/2 

633/8 

14 

221/2 

137/8 

13/4 

43/8 

I63/i6 

66 

77 

27/16 

71/2 

69l/2 

15 

23.1/2 

147/8 

1  13/16 

41/2 

171/4 

72 

831/s 

27/16 

71/2 

755/8 

16 

25 

15  7/8  j 

17/8 

43/4 

181/2 

The  Rockwood  Pipe  Joint.  —  Tfle  system  of  flanged  joints  now  in 
common  use  for  high  pressures,  made  by  slipping  a  flange  over  the  pipe, 
expanding  the  end  of  the  pipe  by  rolling  or  peening,  and  then  facing  it  in 
a  lathe,  so  that  when  the  flanges  of  two  pipes  are  bolted  together  the 
bearing  of  the  joint  is  on  the  ends  of  the  pipes  themselves  and  not  on  the 
flanges,  was  patented  by  George  I.  Rockwood,  April  5,  1897,  No.  580,058, 
and  first  described  in  Eng.  Rec.,  July  20,  1895.  The  joint  as  made  by 
different  manufacturers  is  known  by  various  trade  names,  as  Walmanco, 
Van  Stone,  etc. 

Matheson  Joint  and  Converse  Lock-joint  Pipe. — Sizes,  external 
diameters  2  to  20  in.,  22,  24,  26,  28,  and  30  in.  Kimberley  Joint  Pipe, 
6  to  30  in.  These  pipes  are  considerably  lighter  than  standard  pipe. 
The  Converse  and  Kimberley  joints  are  made  with  special  forms  of  ex- 
ternal hubs,  filled  and  calked  with  lead.  The  Matheson  joint  is  also 
a  lead-packed  joint,  but  the  bell  or  socket  is  made  by  expanding  one  of 
the  pipes,  the  end  being  reinforced  by  a  steel  band.  The  lead  required 
per  joint  is  less  than  for  other  lead- joint  pipes  of  the  same  diameter. 


PIPE   FITTINGS. 

Dimensions  of  Standard  Cast-Iron  Flanged  Pipe  Fittings,  for  Pres- 
sures up  to  125  Lb.  per  Sq.  In.  (Adopted  March  20,  1914,  by  a 
joint  committee  of  manufacturers  and  of  the  Am.  Soc.  M.  E.) 
Dimensions  in  the  tables,  pages  213  and  214,  refer  t9  corresponding 
letters  on  the  sketches  on  page  215.  For  dimensions  of  flanges 
and  bolts  see  Table  of  Standard  Flanges,  pages  209  and  210. 


213 
Standard  Cast  Iron  Flanged  Pipe  Fittings  for  Pressures  up  to  125  lb. 

per  Sq.  In.  (see  sketches  p.  215.) 


Size. 

Tees,  Crosses 
and  Ells. 

Long 
Radius 
Ells. 

45 
degree 
Ells. 

Laterals. 

Re- 
ducers. 

Min. 
Thick- 
ness of 
Metal. 

H/4 

j* 

f>/2 

3./2 

41/2 

6 
7 

8 
9 
10 
12 
14 
15 
16 
18 
20 
22 
24 
26 
28 
30 
32 
34 
36 
38 
40 
42 
44 
46 
48 
50 
52 
54 
56 
58 
60 
62 
64 
66 
68 
70 
72 
74 
76 
78 
80 
82 
84 
86 
88 
90 
92 
94 
96 
98 
100 

A-A 

fy 

9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
20 
22 
24 
28 
29 
30 
33 
36 
40 
44 
46 
48 
50 
52 
54 
56 
58 
60 
62 
64 
66 
68 
70 
74 
78 
82 
84 
88 
90 
94 
96 

too 

102 
106 
108 
112 
116 
118 
120 
124 
126 
130 
134 
136 
138 
142 
146 
148 

A 

31/2 
33/4 

4l/2 

51/2 

6 

61/2 
71/2 

8 

81/2 

9 
10 
11 
12 

14 

g.A 

161/2 

18 
20 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
37 
39 
41 
42 
44 
45 
47 
48 
50 
51 
53 
54 
56 
58 
59 
60 
62 
63 
65 
67 
68 
69 
71 
73 
74 

B 
5 

51/2 
61/2 

73/4 
81/2 

91/2 
101/4 
11  V2 
123/4 
14 
151/4 
161/2 

19 

21  1/2 
223/4 

261/2 

IV/2 
36l/2 
39 

4H/2 

44 

£'/' 
5siI/2 

56* 

61  1/2 
64 

6V/2 
Jl  A 

76l/2 
79 

8H/2 

84 

86  1/2 
89 

9H/2 

94 
96l/2 
99 

101  1/2 

104 
1061/2 
109  . 

!!11/2 

1161/2 

119 

!ir/2 

!£'/2 

C 

]3/4 

21/4 
21/2 

3 

31/2 

4 

41/2 

51/2 
51/2 

6 

61/2 

71/2 
71/2 

8 
8 

81/2 
91/2 

10 
11 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

D 

JV, 
}|V, 

13 

Hl/2 

15 
|5./2 

18 

g* 

24 
251/2 
30 
33 

341/2 

*>/, 

43 
46 
491/2 
53 
56 
59 

E 

53/4 

61/4 

8 

,r/2 
a* 

12i/2 

131/2 
141/2 
161/2 
171/2 
191/2 

201/2 
2?./, 

281/2 
30 
32 
35 
371/2 
401/2 
44 
461/2 
49 

F 

13/4 
13/4 

21/2 
21/2 

3 
3 
3 

31/2 
31/2 

41/2 
41/2 

51/2 

6 
6 

61/2 

8 

81/2 

9 
9 

91/2 

10 

G 
"6  " 

61/2 

J.A 

9 
10 
11 

M'/2 

14 
16 
17 
18 
19 
20 
22 
24 
26 
28 
30 
32 
34 
36 
38 
40 
42 
44 
46 
48 
50 
52 
54 
56 
58 
60 
62 
64 
66 
68 
70 
72 
74 
76 
78 
80 
82 
84 
86 
88 
90 
92 
94 
% 
98 
100 

7/16 

l^ 
7/16 

MM 

7/16 

!/!« 
7/16 
1/2 
1/2 
1/2 
9/16 
5/8 
5/8 

$' 

%16 

7/8 

1/16 
1/8 
3/16 

1/4 
5/16 

y? 

7/16 
1/2 
9/16 
5/8 

"/" 

8r 

'     15/16 

21/16 
21/8 
23/16 
21/4 
25/16 
27/i6 
21/2 
29/16 
25/8 
2  H/16 
23/4 
2  13/16 
27/8 
2  15/16 

31/16 
31/8 
33/is 
31/4 
35/ia 
33/8 
31/2 
39/16 
35/8 
3  11/16 
33/4 

214 


MATERIALS. 


Dimensions  of  American  Standard  Flanged  Reducing  Fittings.     Short 
Body  Pattern.      (All  Dimensions  in  Inches.) 

Long  body  patterns  are  used  when  outlets  are  larger  than  those  in 
table,  and  have  the  same  dimensions  as  straight  size  fittings.  All  re- 
ducing fittings  from  1  to  16  in.  inclusive  have  same  dimensions  as 
straight  size  fittings.  The  dimensions  of  reducing  fittings  are  always 
regulated  by  the  reduction  of  the  outlet. 


18 
20 
22 
24 
26 
28 
30 
32 
34 
56 
38 
40 
42 
44 
46 
48 
50 
52 
54 
56 
58 

Tees,  Ells,  Crosses. 

Laterals. 

i 

w 

~60~ 
62 
64 
66 
68 
70 
72 
74 
76 
78 
80 
82 
84 
86 
88 
90 
92 
94 
96 
98 
100 

Tees,  Ells, 
and  Crosses. 

J* 

Cfl+3 

*3 

S-o 

12 
14 
15 
16 
18 
18 
20 
20 
22 
24 
24 
26 
28 
28 
30 
32 
32 
34 
36 
36 
38 

AA 

26 
28 
28 
30 
32 
32 
36 
36 
38 
40 
40 
44 
46 
46 
48 
52 
52 
54 
58 
58 
62 

A 

13 
14 
14 
15 
16 
16 
18 
18 
19 
20 
20 
22 
23 
23 
24 
26 
26 
27 
29 
29 
31 

B 

N  "£ 
W^ 

So 

a* 

9 
10 
10 
12 
12 
14 
15 

D 

26 
28 
29 
32 
35 
37 
39 

E 

F 

1 

V2 

o'/2 

0 
0 

H 

271/2 
291/2 
31  1/2 
34  i/2 
38 
40 
42 

S4^ 

w5 

& 

S-8 

40 
40 
42 
44 
44 
46 
48 
48 
50 
52 
52 
54 
56 
56 
58 
60 
60 
62 
64 
64 
66 

AA 

66 
66 
68 
70 
70 
74 
80 
80 
84 
86 
86 
88 
94 
94 
96 
100 
100 
104 
106 
106 
110 

A 

~33 
33 
34 
35 
35 
37 
40 
40 
42 
43 
43 
44 
47 
47 
48 
50 
50 
52 
53 
53 
55 

B 

"4? 

42 
44 
45 
46 
47 
48 
49 
50 
52 
53 
54 
56 
57 
58 
61 
62 
63 
64 
65 
67 

17  l/2 

18 
19 
20 

11 

24 
25 
26 
28 
29 
30 
31 
33 
34 
35 
36 
37 
39 
40 

25 
27 

281/2 

\Vh 

37 
39 

Extra  Heavy  American  Standard  Flanged  Reducing  Fittings. 
Body  Pattern.      (All  Dimensions  in  Inches.) 


Short 


1 

CO 

Tees,  Ells  and  Crosses. 

Laterals. 

| 

c/5 

Tees,  Ells  and  Crosses. 

ti 

05^3 

go 

^"o 

AA 

A 

K 

S-^ 

.3   Q) 
C/253 

So 

&£ 

D 

E 

F 

H 

II 

a* 

§3 

AA 

A 

K 

18 
20 
22 
24 
26 
28 
30 
32 

12 
14 
15 
16 
18 
18 
20 
20 

28 
31 
33 
34 
38 
38 
41 
41 

14 

151/2 
161/2 

17 
19 
19 

20  l/2 
201/2 

17 

1SV2 
gv, 

24 

251/2 
26i/2 

9 
10 
10 
12 

34 
37 
40 
44 

31 
34 
37 
41 

3 
3 
3 
3 

8'" 

39 

43 

34 
36 
38 
40 
42 
44 
46 
48 

22 
24 
24 
26 
28 
28 
30 
32 

44 
47 
47 
50 
53 
53 
55 
58 

22 

231/2 

•§>* 

2$ 
i172 

28 
291/2 
301/2 

31  1/2 
331/2 

341/2 

351/2 
37i/2 

Standard  Brass  Flanges  as  adopted  Sept.  17,  1913,  by  the  Committee 
of  manufacturers  on  the  standardization  of  Valves  and  Fittings,  to  be- 
come effective  Jan.  1,  1914  are  listed  on  page  215.  The  bolt  holes  for 
these  flanges  are  to  be  drilled  i/ie  in.  greater  than  the  bolt  diameter  for 
sizes  2  in.  and  smaller,  and  %  in.  greater  than  the  bolt  diameter  for 
sizes  2l/2  in.  and  larger.  The  flanges  have  smooth,  plain  faces,  and  when 
coupled  to  extra  heavy  iron  flanges,  the  latter  should  have  the  raised 
surface  faced  off. 


STANDARD   BRASS  FLANGES. 


215 


Side  Outlet 

Tee 
STEAIGHT  SIZE  FITTINGS. 

J±H      'M**: 


Laterals 
REDUCING  FITTINGS. 


Jfoducers 


The  dimensions  on  these  sketches  refer  to  the  corresponding  letters 
in  the  tables  of  flanged  fittings,  pages  213  and  214,  and  also  to  the 
reference  letters  in  the  tables  of  screwed  fittings,  page  216. 

Standard  Brass  Flanges. 


Standard  —  For  Pressures  up 
to  125  Lb. 

Extra  Heavy  —  For  Pressures 
up  to  250  Lb. 

Size, 
In. 

Diam., 
In. 

Thick- 
ness, 
In. 

Bolt 
Circle  , 
In. 

No. 
of 
Bolts. 

Size 
of 
Bolts, 
In. 

Diam., 
In. 

Thick- 
ness, 
In. 

Bolt 
Circle, 
In. 

No. 
of 
Bolts. 

Size 
of 
Bolts, 
In. 

V4&3/8 

2V2 

9/32 

1  n/16 

4 

3/8 

3 

3/8 

2 

4 

7/16 

1/2 

3/4 

3 

3V2 

5/16 
H/32 

2V8 
2V2 

4 
4 

3/8 
3/8 

jv, 

13/32 
7/16 

23/8 

27/8 

4 
4 

'£ 

1 

4 

3/8 

3 

4 

7/16 

41/2 

V2 

3V4 

4 

V2 

H/4 

4V2 

1V32 

33/8 

4 

7/16 

5 

n/32 

33/4 

4 

V2 

H/2 

5 

Vl6 

37/8 

4 

V2 

6 

9/16 

4V2 

4 

5/8 

2 

6 

V2 

43/4 

4 

5/8 

6V2 

5/8 

5 

4 

5/8 

2V2 

7 

9/16 

5V2 

4 

5/8 

7V2 

n/16 

57/8 

4 

3/4 

3 

?V2 

5/8 

6 

4 

'5/8 

8V4 

3/4 

6Vs 

8 

3/4 

3V2 

81/2 

H/16 

7 

4 

5/8 

9 

13/16 

7V4 

8 

3/4 

4 

9 

n/16 

7V2 

8 

5/8 

10 

7/8 

7V8 

8 

3/4 

4V2 

9V4 

23/32 

73/4 

8 

3/4 

10V2 

7/8 

8V2 

8 

3/4 

5 

10 

3/4 

8V2 

8 

3/4 

11 

15/16 

9V4 

8 

3/4 

6 

11 

.13/16 

9i/2 

8 

3/4 

12V2 

105/8 

12 

3/4 

7 

12V2 

7/8 

10:V4 

8 

3/4 

14 

Vl6 

11  7/8 

12 

7/8 

8 

131/2 

16/16 

11  3/4 

8 

3/4 

15 

1/8 

13 

12 

7/8 

9 

15 

15/16 

131/4 

12 

3/4 

16i/4 

1/8 

14 

12 

1 

10 

16 

141/4 

12 

7/8 

17V2 

3/16 

15V4 

16 

1 

12 

19 

1Vl6 

17 

12 

7/8 

201/2 

V4 

173/4 

16 

IV8 

'216 


MATERIALS. 


Dimensions  of  Screwed  Cast  Iron  and  Malleable  Pipe  Fittings,  For 
Steam  and  Water.      (Crane  Co.,  Chicago,  1914.) 

R  =  regular  fitting;  E.H.  ~  extra  heavy  fitting.     For  meaning  of 
dimensions  see  sketches  p.  215.     Dimensions  in  inches. 


. 

Long 

51*"      Tee,  Cross,  Ell. 

Rad. 

45  Deg.  Ell. 

Lateral. 

Reducer.* 

wng. 

Ell. 

Dimension.       A 

B 

C 

D 

E 

G 

Size, 
Ins. 

Cast  Iron. 

Mall. 

Mall. 

Cast  Iron. 

Mall. 

C.I. 

C.I. 

C.I. 

Mall. 

R. 

E.  H. 

E.H. 

E.  H. 

R. 

E.H. 

E.H. 

R. 

R. 

R. 

E.  H. 

1/4 

13/16 

1  i/ifi 

3/4 

3/4 

3/8 

15/ie 

1  1/4 

13/16 

7/8 

1/2 

1  1/8 

1  1/2 

7/8 

1    ' 

21/2 

1  7/8 

3/4 

15/16 

13/4 

1 

U/8 

3 

21/4 

1  H/16 

1 

1  7/16 

2" 

2 

2  1/2 

U/8 

13/8" 

15/16 

31/2 

23/4 

2 

1   1/4 

13/4 

21/4 

21/4 

3 

15/16 

H/2 

41/4 

31/4 

21/8  ' 

23/8 

1   V2 

1  15/16 

29/16 

31/2 

17/16 

15/8 

1  H/16 

47/8 

313/16 

21/4 

2  H/16 

2 

21/4 

3 

3 

4 

1  15/16 

2 

53/4 

41/2 

27/16 

23/16 

2l/2 

2  H/16 

31/2 

31/2 

43/4 

1  IS/16 

21/4 

21/4 

61/4 

53/16 

2  H/16 

3 

31/8 

41/8 

41/8 

51/2 

2  3/16 

21/2 

21/2 

77/8 

61/8 

2  15/16 

3l/2 

37/16 
33/4 

4H/16 
51/8 

45/8 
51/8 

61/423/8 
7         25/8 

"29/ie 
23/4 

25/8 
2  13/16 

87/8 
93/4 

67/8 
75/8 

31/8 
33/8 

41/2 

41/16 

51/2 

55/8 

73/4!213/i63 

115/8 

91/4 

35/8 

5 

47/16 

61/8 

61/4 

31/16    35/ie 

115/8 

91/t 

37/8 

6 

51/8 

71/4 

71/4 

91/2 

37/16  133/4 

137/16 

103/4 

43/8 

7 

5  13/16 

81/8 

37/8 

4 

151/4 

121/4 

4  13/16 

8 

61/2 

91/8 

41/4 

43/4 

1615/i6 

135/8 

51/4 

9 
10 

73/16 
77/8 

4H/16 
53/16 

47/8  ' 

20H/16 

163/4 
163/4 

5  H/16 
63/ie 

12 

91/4 

133/8 

6 

51/2 

1 

195/s 

71/8 

*  The  reducers  are  for  reducing  from  the  size  of  pipe  given  to  the 
next  smaller  size.  In  addition,  malleable  reducers  are  listed  for  1  %  X 
Vi,^lA  X  1,  1  Yi  X  Vi,  2  x  1,  2  X  Vi-  The  dimension  G  given  in  the 
table  is  the  same  for  these  special  fittings  as  for  the  regular  fittings 
given  above. 

Strength  of  Pipe  Fittings. — To  determine  the  actual  bursting  strength 
of  cast  iron  fittings,  and  also  to  deter